A104989 - OEIS

%I #6 Jun 08 2021 15:29:58

%S 1,3,13,69,433,3133,25657,234537,2367825,26176981,314670353,

%T 4088360569,57112939433,853922061413,13609089281849,230346936181465,

%U 4127180489763649,78046835384582069,1553536327234953153

%N Row sums of triangle A104988, which equals the matrix square of triangle A104980.

%H G. C. Greubel, <a href="/A104989/b104989.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = Sum_{k=0..n} A104988(n, k).

%t nmax:=30;

%t T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*T[n, k+1] + Sum[T[j, 0]*T[n, j+k+1], {j, 0, n-k-1}]]]]; (* T=A104980 *)

%t M:= M= With[{q = nmax}, Table[If[j>i, 0, T[i,j]], {i,0,q}, {j,0,q}]];

%t f:= f= MatrixPower[M, 2];

%t a[n_]:= a[n]= Sum[f[[n+1, k+1]], {k,0,n}];

%t Table[a[n], {n, 0, nmax}] (* _G. C. Greubel_, Jun 08 2021 *)

%o (PARI) {a(n)=if(n<0,0,sum(k=0,n,(matrix(n+1,n+1,m,j,if(m==j,1,if(m==j+1,-m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m),m-j-1))))^-2)[n+1,k+1]))}

%Y Cf. A104980, A104988.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 10 2005