A104987 Row sums of triangle A104986, which equals the matrix logarithm of triangle A104980.

0, 1, 4, 14, 58, 300, 1886, 13878, 116310, 1090500, 11296810, 128102714, 1578342010, 20998804576, 300081098918, 4584908039142, 74594230462318, 1287634918033836, 23506502407089874, 452508152936326482

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..200

Mathematica

(* First program *)
nmax = 19;
M = Table[If[n==k, 0, If[n==k+1, -n+1, -Coefficient[(1 -1/Sum[i!*x^i, {i, 0, n}])/x + O[x]^n, x, n-k-1]]], {n, 1, nmax+1}, {k, 1, nmax+1}];
T[n_, k_]/; 0<=k<=n:= Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p, n+1}]; T[_, _] = 0;
a[n_]:= Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 09 2018, from PARI *)
(* Second program *)
t[n_, k_]:= t[n, k] = If[n<k || k<0, 0, If[n==k, 1, If[n==k+1, n, k*t[n, k+1] + Sum[t[j, 0]*t[n, j+k+1], {j, 0, n-k-1}]]]];
M:= M= With[{q=52}, Table[If[k>=n, 0, t[n, k]], {n, 0, q}, {k, 0, q}]];
f[j_]:= f[j]= MatrixPower[M, j];
T[n_, k_]:= T[n, k]= If[k>n-1, 0, Sum[(-1)^(j-1)*f[j][[n+1, k+1]]/j, {j, n}]];
a[n_]:= a[n]= Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jun 08 2021 *)

Prog

(PARI) {a(n)=sum(k=0, n, sum(p=1, n+1, (-1)^p*(matrix(n+1, n+1, m, j, if(m==j, 0, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^p)[n+1, k+1]/p))}

Crossrefs

Cf. A104986, A104980.

Sequence in context: A096242 A360198 A296943 * A149492 A307488 A351634

Adjacent sequences: A104984 A104985 A104986 * A104988 A104989 A104990

Keyword

nonn

Author

Paul D. Hanna, Apr 10 2005

Status

approved