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.
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 24 2007
STATUS
approved