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A132616 Column 0 of triangle A132615.

%I A132616 #14 Apr 12 2020 21:41:32
%S A132616 1,1,1,6,80,1666,47232,1694704,73552752,3744491970,218684051648,
%T A132616 14406896813608,1056681951098592,85379764462169382,
%U A132616 7534286318509305600,720884741940337283712,74330131862002429961712,8215901579822006354547330,969069489665924620416715008
%N A132616 Column 0 of triangle A132615.
%C A132616 Triangle T = A132615 is generated by odd matrix powers of itself such that row n+1 of T = row n of T^(2n-1) with appended '1' for n >= 0 with T(0,0) = 1.
%F A132616 From _Benedict W. J. Irwin_, Nov 29 2016: (Start)
%F A132616 Conjecture: a(n) is described by a series of nested sums,
%F A132616 a(2) = Sum_{i=1..1} 1,
%F A132616 a(3) = Sum_{i=1..1+2} Sum_{j=1..i} 1,
%F A132616 a(4) = Sum_{i=1..1+4} Sum_{j=1..i+2} Sum_{k=1..j} 1,
%F A132616 a(5) = Sum_{i=1..1+6} Sum_{j=1..i+4} Sum_{k=1..j+2} Sum_{l=1..k} 1,
%F A132616 and so on, where 2, 4, 6,... are the even numbers. (End)
%e A132616 G.f. = 1 + x + x^2 + 6*x^3 + 80*x^4 + 1666*x^5 + 47232*x^6 + ...
%t A132616 a[ n_, k_: 1] := a[n, k] = If[ n < 2, Boole[n >= 0], Sum[ a[n - 1, i], {i, k + 2 (n - 2)}]]; (* _Michael Somos_, Nov 29 2016 *)
%o A132616 (PARI) {a(n) = my(A = vector(n+1), p); A[1] = 1; for(j=1, n-1, p = (n-1)*(n-2) - (n-j-1)*(n-j-2); A = Vec((Polrev(A) + x * O(x^p)) / (1-x))); A = Vec((Polrev(A) + x * O(x^p)) / (1-x)); A[p+1]}
%o A132616 (PARI) {a(n, k=1) = if( n<2, n>=0, sum(i=1, k + 2*n-4, a(n-1, i)))}; /* _Michael Somos_, Nov 29 2016 */
%Y A132616 Cf. A132615 (triangle).
%Y A132616 Other columns: A132617, A132618, A132619.
%K A132616 nonn
%O A132616 0,4
%A A132616 _Paul D. Hanna_, Aug 24 2007