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

%I A104987 #11 Jun 09 2021 02:27:55
%S A104987 0,1,4,14,58,300,1886,13878,116310,1090500,11296810,128102714,
%T A104987 1578342010,20998804576,300081098918,4584908039142,74594230462318,
%U A104987 1287634918033836,23506502407089874,452508152936326482
%N A104987 Row sums of triangle A104986, which equals the matrix logarithm of triangle A104980.
%H A104987 G. C. Greubel, <a href="/A104987/b104987.txt">Table of n, a(n) for n = 0..200</a>
%t A104987 (* First program *)
%t A104987 nmax = 19;
%t A104987 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 A104987 T[n_, k_]/; 0<=k<=n:= Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p,n+1}]; T[_, _] = 0;
%t A104987 a[n_]:= Sum[T[n, k], {k,0,n}];
%t A104987 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Aug 09 2018, from PARI *)
%t A104987 (* Second program *)
%t A104987 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}]]]];
%t A104987 M:= M= With[{q=52}, Table[If[k>=n, 0, t[n, k]], {n,0,q}, {k,0,q}]];
%t A104987 f[j_]:= f[j]= MatrixPower[M, j];
%t A104987 T[n_, k_]:= T[n, k]= If[k>n-1, 0, Sum[(-1)^(j-1)*f[j][[n+1, k+1]]/j, {j, n}]];
%t A104987 a[n_]:= a[n]= Sum[T[n, k], {k,0,n}];
%t A104987 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Jun 08 2021 *)
%o A104987 (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))}
%Y A104987 Cf. A104986, A104980.
%K A104987 nonn
%O A104987 0,3
%A A104987 _Paul D. Hanna_, Apr 10 2005