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

Author

Paul D. Hanna, Apr 10 2005

Keywords

• nonn

Links

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

Crossrefs

Cf. A104986, A104980.

Programs

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=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 *)

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))}