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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2015
STATUS
approved