A264416 G.f.: Sum_{n>=0} x^n * (d/dx)^(n^2) x^(n^2) * (1+x)^n / (n^2)!, where (d/dx)^k denotes the k-th derivative operator.

1, 1, 3, 11, 46, 234, 1269, 7609, 48501, 328831, 2347655, 17551462, 136902440, 1109043889, 9308756773, 80704240807, 721319484280, 6632169860334, 62631220229804, 606525577239266, 6015393685594488, 61024457251302195, 632560831051022660, 6693134988051996106, 72226722923184020925, 794238562047133369308

Offset

0,3

Comments

Compare to the trivial: Sum_{n>=0} x^n * (d/dx)^(n^2) x^(n^2) / (n^2)! = 1/(1-x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..165

Formula

a(n) = Sum_{k=0..[n/2]} binomial(n-k, k) * binomial((n-k)^2 + k, k).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 234*x^5 + 1269*x^6 + 7609*x^7 + 48501*x^8 + 328831*x^9 + 2347655*x^10 +...
where
A(x) = 1 + x * d/dx x*(1+x) + x^2 * d^4/dx^4 x^4*(1+x)^2/4! + x^3 * d^9/dx^9 x^9*(1+x)^3/9! + x^4 * d^16/dx^16 x^16*(1+x)^4/16! + x^5 * d^25/dx^25 x^25*(1+x)^5/25! + x^6 * d^36/dx^36 x^36*(1+x)^6/36! +...

Prog

(PARI) {Dx(n, F)=my(G=F); for(i=1, n, G=deriv(G)); G}
{a(n) = my(A); A = sum(m=0, n, x^m * Dx(m^2, x^(m^2)*(1+x +x*O(x^n))^m)/(m^2)! ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial((n-k)^2+k, k))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A129579 A030814 A030976 * A290712 A248208 A112567

Adjacent sequences: A264413 A264414 A264415 * A264417 A264418 A264419

Keyword

nonn

Author

Paul D. Hanna, Nov 17 2015

Status

approved