A184355 G.f.: Sum_{n>=0} x^n / [Sum_{k>=0} C(n+k-1,k)^2*(-x)^k]^n.

1, 1, 2, 9, 58, 515, 6117, 89015, 1582412, 33346657, 816759195, 22980062928, 733407740393, 26280164467356, 1048777166376622, 46274709506560769, 2242998363098170350, 118779992677414890453, 6838446632281205146327, 426147448479639862008434, 28631211803547719170094520

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..n-1} C(n-1,k)^2*(-x)^k]^n.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 58*x^4 + 515*x^5 + 6117*x^6 +...
which equals the sum of the series:
A(x) = 1 + x/(1 - x + x^2 - x^3 + x^4 - x^5 +...)
+ x^2/(1 - 2^2*x + 3^2*x^2 - 4^2*x^3 + 5^2*x^4 + 6^2*x^5 +...)^2
+ x^3/(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)^3
+ x^4/(1 - 4^2*x + 10^2*x^2 - 20^2*x^3 + 35^2*x^4 - 56^2*x^5 +...)^4
+ x^5/(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)^5
+ x^6/(1 - 6^2*x + 21^2*x^2 - 56^2*x^3 + 126^2*x^4 - 252^2*x^5 +...)^6 +...
The g.f. can also be expressed as:
A(x) = 1 + x*(1+x) + x^2*(1+x)^6/(1-x)^2
+ x^3*(1+x)^15/(1 - 2^2*x + x^2)^3
+ x^4*(1+x)^28/(1 - 3^2*x + 3^2*x^2 - x^3)^4
+ x^5*(1+x)^45/(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)^5
+ x^6*(1+x)^66/(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)^6
+ x^7*(1+x)^91/(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)^7 +...

Prog

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

Crossrefs

Cf. A184356, A183165, A183166.

Sequence in context: A191811 A005364 A267464 * A346519 A151616 A199543

Adjacent sequences: A184352 A184353 A184354 * A184356 A184357 A184358

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2011

Status

approved