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\nusing namespace std;\n\nconst int64_t MOD = 1000000007;\n\n// O(mn) m+1項間漸化式のn要素目までの和を前から順に項を計算し足していくことで求める\nint64_t simple_sum(const vector& coefficients, const vector& firsts, int64_t n);\n\n// O(m^3 log(n)) m+1項間漸化式のn要素目までの和を累乗和で求める\nint64_t power_series_sum(const vector& coefficients, const vector& firsts, int64_t n);\n\n// O(m^2 log(n)) m+1項間漸化式のn要素目までの和を総和の漸化式+kitamasa法で求める（今回のメイン）\nint64_t kitamasa_sum(const vector& coefficients, const vector& firsts, int64_t n);\n\n\nint dummy = 0;\n// 時間計測関数\ntemplate \nauto timeit(F f, string func_name, int num_exec, const vector& coefficients, const vector& firsts, int64_t n) {\n\tauto start = std::chrono::system_clock::now();\n\t\n\tfor (int i = 0; i < num_exec; i++) {\n\t\tdummy = f(coefficients, firsts, n);\n\t}\n\t\n\tauto end = std::chrono::system_clock::now();\n auto dur = end - start;\n auto usec = std::chrono::duration_cast(dur).count();\n \n cout << func_name << \": \" << (usec / (double)num_exec / 1000) << \" ms\" << endl;\n}\n\n\nint main() {\n\tvector firsts(100); // 初項 a_1, a_2, ... , a_m\n\tvector coefficients(100); // 係数 c_1, c_2, ..., c_m\n\tfor (int i = 0; i < (int)firsts.size(); i++) {\n\t\tfirsts[i] = i + 1;\n\t\tcoefficients[i] = i + 1;\n\t}\n\t\n\tint64_t n = 100000000; // n項目までの和を求める\n\t\n\tint num_exec = 10; // 計測回数\n\t\n\t//timeit(simple_sum, \"simple_sum\", num_exec, coefficients, firsts, n);\n\ttimeit(power_series_sum, \"power_series_sum\", num_exec, coefficients, firsts, n);\n\ttimeit(kitamasa_sum, \"kitamasa_sum\", num_exec, coefficients, firsts, n);\n}\n\n\n/*\n\n----------------------------------------------------------\n\n今回のメイン\n\nm+1項間漸化式を総和の漸化式(m+2項間)に変形\n\n*/\n\n// O(m) 総和の漸化式の係数を求める\nvector coefficients_sums(const vector& coefficients) {\n\t\n\tint m = coefficients.size();\n\tvector new_coefficients(m + 1);\n\t\n\tnew_coefficients[0] = -coefficients[0];\n\tfor (int i = 0; i + 1 < m; i++) {\n\t\tnew_coefficients[i + 1] = coefficients[i] - coefficients[i + 1];\n\t}\n\tnew_coefficients[m] = coefficients[m - 1] + 1;\n\t\n\treturn new_coefficients;\n}\n\n// O(m) m+1項間漸化式の初項からm+1項目までを求め、配列で返す\nvector first_sums(const vector& coefficients, const vector& firsts) {\n\t\n\tint m = firsts.size();\n\tvector sums(m + 1);\n\tsums[0] = firsts[0];\n\tfor (int i = 1; i < m; i++) {\n\t\tsums[i] = (firsts[i] + sums[i - 1]) % MOD;\n\t}\n\t\n\tint64_t next = 0;\n\tfor (int i = 0; i < m; i++) {\n\t\tnext = (next + (coefficients[i] * firsts[i])) % MOD;\n\t}\n\tsums[m] = (next + sums[m - 1]) % MOD;\n\t\n\treturn sums;\n}\n\n// O(m^2 log(n)) m+1項間漸化式のn要素目をkitamasa法で求める\nint64_t kitamasa(const vector& coefficients, const vector& firsts, int64_t n);\n\nint64_t kitamasa_sum(const vector& coefficients, const vector& firsts, int64_t n) {\n\tauto a_sums = first_sums(coefficients, firsts);\n\tauto c_sums = coefficients_sums(coefficients);\n\treturn kitamasa(c_sums, a_sums, n);\n}\n\n/*\n\n----------------------------------------------------------\n\n単純に前から計算\n\n*/\n\n// coefficients : m+1項間漸化式の係数c_1 ~ c_mを与える\n// firsts : m+1項間漸化式の初項a_1 ~ a_mを与える\n// n : n項目までの和を求める\nint64_t simple_sum(const vector& coefficients, const vector& firsts, int64_t n) {\n\t\n\tint m = firsts.size();\n\tdeque nexts(m);\n\tint64_t sum = 0;\n\tfor (int i = 0; i < m; i++) {\n\t\tsum = (sum + firsts[i]) % MOD;\n\t\tnexts[i] = firsts[i];\n\t}\n\t\n\tfor (int i = m; i < n; i++) {\n\t\tint64_t next = 0;\n\t\tfor (int j = 0; j < m; j++) {\n\t\t\tnext = (next + coefficients[j] * nexts[j]) % MOD;\n\t\t}\n\t\t\n\t\tsum = (sum + next) % MOD;\n\t\tnexts.pop_front();\n\t\tnexts.push_back(next);\n\t}\n\t\n\treturn sum;\n}\n\n\n/*\n\n----------------------------------------------------------\n\n累乗和\n\n出典：\nプログラミングコンテストチャレンジブック第2版\nP180 ~ P185\n一部改変\n\n*/\n\nusing vec = vector;\nusing mat = vector>;\nmat mul(mat &A, mat &B) {\n\tmat C(A.size(), vec(B[0].size()));\n\tfor (int i = 0; i < (int)A.size(); i++) {\n\t\tfor (int k = 0; k < (int)B.size(); k++) {\n\t\t\tfor (int j = 0; j < (int)B[0].size(); j++) {\n\t\t\t\tC[i][j] = (C[i][j] + A[i][k] * B[k][j]) % MOD;\n\t\t\t}\n\t\t}\n\t}\n\t\n\treturn C;\n}\n\nmat pow(mat A, int64_t n) {\n\tmat B(A.size(), vec(A.size()));\n\tfor (int i = 0; i < (int)A.size(); i++) {\n\t\tB[i][i] = 1;\n\t}\n\twhile (n > 0) {\n\t\tif (n & 1) B = mul(B, A);\n\t\tA = mul(A, A);\n\t\tn >>= 1;\n\t}\n\t\n\treturn B;\n}\n\n\nmat make_companion(const vector& coefficients) {\n\tint m = coefficients.size();\n\tmat A(m, vec(m));\n\tfor (int i = 0; i < m; i++) {\n\t\tA[0][i] = coefficients[m - i - 1];\n\t}\n\tfor (int i = 0; i < m - 1; i++) {\n\t\tA[1 + i][i] = 1;\n\t}\n\t\n\treturn A;\n}\n\nmat matrix_power_series(const vector& coefficients, int n) {\n\tint m = coefficients.size();\n\tauto A = make_companion(coefficients);\n\tmat B(m * 2, vec(m * 2));\n\tfor (int i = 0; i < m; i++) {\n\t\tfor (int j = 0; j < m; j++) {\n\t\t\tB[i][j] = A[i][j];\n\t\t}\n\t\tB[m + i][i] = B[m + i][m + i] = 1;\n\t}\n\t\n\tB = pow(B, n);\n\t\n\tmat S(m, vec(m));\n\tfor (int i = 0; i < m; i++) {\n\t\tfor (int j = 0; j < m; j++) {\n\t\t\tS[i][j] = B[m + i][j];\n\t\t}\n\t}\n\t\n\treturn S;\n}\n\nint64_t power_series_sum(const vector& coefficients, const vector& firsts, int64_t n) {\n\tauto S = matrix_power_series(coefficients, n);\n\tint m = firsts.size();\n\tint64_t sum = 0;\n\tfor (int i = 0; i < m; i++) {\n\t\tsum = (sum + S[m - 1][i] * firsts[m - i - 1]) % MOD;\n\t}\n\treturn sum;\n}\n\n/*\n\n----------------------------------------------------------\n\nkitamasa法\n\n出典：\nm項間漸化式の高速なアルゴリズム - 競技プログラミングをするんだよ\nhttp://nitcoder000.hatenablog.com/entry/kitamasa\n一部改変\n\n*/\n\n#define reE(i,a,b) for(auto i=(a);(i)<=(b);i++)\n#define rE(i,b) reE(i,0,b)\n#define reT(i,a,b) for(auto i=(a);(i)<(b);i++)\n#define rT(i,b) reT(i,0,b)\n#define rep(i,a,b) reE(i,a,b);\n#define rev(i,a,b) for(auto i=(b)-1;(i)>=(a);(i)--)\n#define itr(i,b) for(auto i=(b).begin();(i)!=(b).end();++(i))\n#define LL int64_t\n#define all(b) (b).begin(),(b).end()\n\n\n/*\n使い方\nM項間漸化式のN項目を計算する。Nの最小は1。\na[N]=sum(c[i]*a[N-M+i-1])の形。\nMrが本体、Xは使わなくてもできる問題はある。(MODを使わないint,LLなど)\nXは半環を満たすものならなんでもよい、+,*のオーバーロードにMODをねじこむ。\nコンストラクタに初項、係数、M,*の単位元、+の単位元の順で引数を与える。\nあとはcalcにNを与えるだけ。\n*/\n#define MAX_LOGN 32\ntemplate \nstruct Mr{\n\tvector first;\n\tvector C;\n\tvector> bin;\n\tT zero,one;\n\tint M;\n\t//n(1,,,2M)をn(1,,,M)に修正、O(M^2)\n\tvoid form(vector &n){\n\t\trev(i, M + 1, 2 * M + 1){\n\t\t\treE(j, 1, M)n[i - j] = (n[i - j] + (C[M - j] * n[i]));\n\t\t\tn[i] = zero;\n\t\t}\n\t}\n\t//lとrを足し合わせる、O(M^2)\n\tvoid add(vector &l, vector &r, vector &ans){\n\t\treE(i, 1, 2 * M)ans[i] = zero;\n\t\treE(i, 1, M)reE(j, 1, M)ans[i + j] = (ans[i + j] + (l[i] * r[j]));\n\t\tform(ans);\n\t}\n\t//初期化、O(M*MAX_LOGN)\n\tMr(const vector& f,const vector& c,int m,T e1,T e0){\n\t\tM = m;\n\t\tfirst.reserve(M + 1);C.reserve(M);\n\t\tzero = e0, one = e1;\n\t\tfirst.push_back(zero); \n\t\trT(i, M){ first.push_back(f[i]); C.push_back(c[i]); }\n\t\tbin.resize(MAX_LOGN);\n\t\trT(i, MAX_LOGN)bin[i].resize(2*M+1);\n\t\trE(i, 2*M)bin[0][i] = zero;\n\t\tbin[0][1] = one;\n\t\treT(i,1, MAX_LOGN){\n\t\t\tadd(bin[i - 1], bin[i - 1], bin[i]);\n\t\t}\n\t}\n\t//N項目の計算、戻り値がTの形であることに注意、O(M^2*logN)\n\tT calc(LL n){\n\t\tn--;\n\t\tvector tmp,result = bin[0];\n\t\tfor (int b = 0; n; b++,n>>=1)\n\t\t\tif (1 & n){ tmp = result; add(tmp, bin[b], result); }\n\t\tT ans = zero;\n\t\treE(i, 1, M)ans = ans + (result[i] * first[i]);\n\t\treturn ans;\n\t}\n};\n\n//テンプレート、デフォルトコンストラクタのオーバーロードを忘れない\nstruct X{\n\tLL val;\n\tX(LL v){ val = v; }\n\tX(){ val = 0; }\n\tLL operator=(const X &another){ return val = another.val; }\n\tLL operator*(const X &another) const {\n\t\tauto tmp = (val*another.val);\n\t\treturn (tmp + (abs(tmp / MOD) + 1) * MOD) % MOD; // 負のmod対策\n\t}\n\tLL operator+(const X &another)const{ return (val+another.val)%MOD; }\n};\n\nint64_t kitamasa(const vector& coefficients, const vector& firsts, int64_t n) {\n\tint m = firsts.size();\n\tvector A, B;\n\tfor (auto a : firsts) A.push_back(X(a));\n\tfor (auto c : coefficients) B.push_back(X(c));\n\tMr mr(A, B, m, X(1), X(0));\n\treturn mr.calc(n).val;\n}\n","codes",[],"stdin","compiler","warning,optimize,boost-1.65.1-gcc-head,gnu++2a,cpp-no-pedantic","createdAt",1517539396,"title","description","isPrivate","compilerInfo",{"_11":59,"_13":59,"_15":59,"_17":59,"_19":3454,"_22":25,"_24":25,"_26":59,"_28":3455},[],[],"results",[3458,3462,3465,3468],{"_31":3459,"_3460":3461},"Control","data","Start",{"_31":3463,"_3460":3464},"StdOut","power_series_sum: 1764.7 ms\nkitamasa_sum: 12.0769 ms\n",{"_31":3466,"_3460":3467},"ExitCode","0",{"_31":3459,"_3460":3469},"Finish","hpplib",[3472,3489,3500,3509,3518,3527,3536,3545,3555,3566,3577,3588,3599,3610,3621,3631],{"_11":3473,"_3474":3475,"_3450":3476,"_3477":3478,"_3479":3480,"_3485":3486},"CLI11","repository","CLIUtils/CLI11","CLI11 is a command line parser for C++11 and beyond 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