A184356 G.f.: Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k]^n.

1, 1, 2, 10, 75, 757, 9955, 161608, 3149491, 72294325, 1919933126, 58189667167, 1991123304634, 76201510956909, 3235630545496281, 151399102211450842, 7760065212106661217, 433404831023513573519, 26253103133315432898270, 1717576707472491422233436, 120912301935843736344714288

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

G.f.: Sum_{n>=0} x^n * (1+x)^(-2*n^2 - n) / [Sum_{k>=0} C(n+k,k)^2*(-x)^k]^n.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 75*x^4 + 757*x^5 + 9955*x^6 +...
equals the sum of the series:
A(x) = 1 + x/(1-x) + x^2/(1 - 2^2*x + x^2)^2 +
+ x^3/(1 - 3^2*x + 3^2*x^2 - x^3)^3
+ x^4/(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)^4
+ x^5/(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)^5
+ x^6/(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)^6 +...
The g.f. can also be expressed as:
A(x) = 1 + x*(1+x)^-3/(1 - 2^2*x + 3^2*x^2 - 4^2*x^3 + 5^2*x^4 -+...)
+ x^2*(1+x)^-10/(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 -+...)^2
+ x^3*(1+x)^-21/(1 - 4^2*x + 10^2*x^2 - 20^2*x^3 + 35^2*x^4 -+...)^3
+ x^4*(1+x)^-36/(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 -+...)^4
+ x^5*(1+x)^-55/(1 - 6^2*x + 21^2*x^2 - 56^2*x^3 + 126^2*x^4 -+...)^5 +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/sum(k=0, m, binomial(m, k)^2*(-x)^k +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*(1+x+x*O(x^n))^(-2*m^2-m)/sum(k=0, n-m+1, binomial(m+k, k)^2*(-x)^k+x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A184355, A183165, A183166.

Sequence in context: A289679 A352270 A136222 * A292631 A295098 A124426

Adjacent sequences: A184353 A184354 A184355 * A184357 A184358 A184359

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2011

Status

approved