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Row sums of triangle A104990, which equals the matrix cube of triangle A104980.
2

%I #7 Jun 08 2021 14:58:28

%S 1,4,22,142,1048,8704,80386,817786,9093340,109794556,1431360958,

%T 20047830262,300343272952,4793871035416,81232799446906,

%U 1456671526257106,27562347560513524,548844246683051860,11474015910364016086

%N Row sums of triangle A104990, which equals the matrix cube of triangle A104980.

%H G. C. Greubel, <a href="/A104991/b104991.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = Sum_{k=0..n} A104990(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, 3];

%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))))^-3)[n+1,k+1]))}

%Y Cf. A104980, A104990.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 10 2005