A132616 Column 0 of triangle A132615.

1, 1, 1, 6, 80, 1666, 47232, 1694704, 73552752, 3744491970, 218684051648, 14406896813608, 1056681951098592, 85379764462169382, 7534286318509305600, 720884741940337283712, 74330131862002429961712, 8215901579822006354547330, 969069489665924620416715008

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..18.

Formula

From Benedict W. J. Irwin, Nov 29 2016: (Start)

Conjecture: a(n) is described by a series of nested sums,

a(2) = Sum_{i=1..1} 1,

a(3) = Sum_{i=1..1+2} Sum_{j=1..i} 1,

a(4) = Sum_{i=1..1+4} Sum_{j=1..i+2} Sum_{k=1..j} 1,

a(5) = Sum_{i=1..1+6} Sum_{j=1..i+4} Sum_{k=1..j+2} Sum_{l=1..k} 1,

and so on, where 2, 4, 6,... are the even numbers. (End)

Example

G.f. = 1 + x + x^2 + 6*x^3 + 80*x^4 + 1666*x^5 + 47232*x^6 + ...

Mathematica

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 *)

Prog

(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]}
(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 */

Crossrefs

Cf. A132615 (triangle).

Other columns: A132617, A132618, A132619.

Sequence in context: A337564 A177776 A349528 * A349657 A323694 A077393

Adjacent sequences: A132613 A132614 A132615 * A132617 A132618 A132619

Keyword

nonn

Author

Paul D. Hanna, Aug 24 2007

Status

approved