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A229113 Row sums of triangle A166890.

%I A229113 #6 Aug 16 2024 18:49:51
%S A229113 1,3,14,115,1474,26306,605538,17116241,574398861,22333212354,
%T A229113 987660842174,48969675370286,2690888441513773,162333780684342177,
%U A229113 10667116965172284025,758434434866124781819,58015602766701416940599,4750972163883049286678765,414721061026426313701858479
%N A229113 Row sums of triangle A166890.
%C A229113 Triangle A166890 transforms diagonals in the table of coefficients of successive iterations of x*(1+x)^2 (cf. A166888).
%e A229113 Triangle A166890 begins:
%e A229113 1;
%e A229113 2, 1;
%e A229113 9, 4, 1;
%e A229113 78, 30, 6, 1;
%e A229113 1038, 364, 63, 8, 1;
%e A229113 18968, 6233, 986, 108, 10, 1;
%e A229113 443595, 139008, 20685, 2072, 165, 12, 1;
%e A229113 12681960, 3833052, 545736, 51494, 3750, 234, 14, 1; ...
%e A229113 of which the row sums form this sequence.
%o A229113 (PARI) {a(n, k)=local(F=x, M, N, P, m=max(n, k), A166890); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); A166890=P~*(N~)^-1;sum(k=0,n,A166890[n+1, k+1])}
%o A229113 for(n=0, 25, print1(a(n), ", "))
%Y A229113 Cf. A166890.
%K A229113 nonn
%O A229113 0,2
%A A229113 _Paul D. Hanna_, Sep 13 2013