A178324 G.f.: A(x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k].

1, 1, 2, 6, 26, 146, 1024, 8776, 89670, 1071174, 14749528, 231333872, 4090080388, 80808196548, 1770893277532, 42769289395020, 1131848169152498, 32655896028179970, 1022576267852973448, 34612454839065740688

Offset

0,3

Comments

Compare the g.f. of this sequence to the identity:

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

Links

Table of n, a(n) for n=0..19.

Formula

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 146*x^5 + 1024*x^6 +...
which equals the series:
A(x) = 1 + x/(1-x) + x^2/(1-4*x+x^2) + x^3/(1-9*x+9*x^2-x^3) + x^4/(1-16*x+36*x^2-16*x^3+x^4) + x^5/(1-25*x+100*x^2-100*x^3+25*x^4-x^5) +...
Also, the g.f. equals the series:
A(x) = 1/((1+x)*(1 - x + x^2 - x^3 +...))
+ x/((1+x)^3*(1 - 2^2*x + 3^2*x^2 - 4^2*x^3 +...))
+ x^2/((1+x)^5*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 +...))
+ x^3/((1+x)^7*(1 - 4^2*x + 10^2*x^2 - 20^2*x^3 +...)) +...

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))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/((1+x+x*O(x^n))^(2*m+1)*sum(k=0, n-m+1, binomial(m+k, k)^2*(-x)^k+x*O(x^n)))), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A107404 A120574 A333044 * A002710 A132609 A168450

Adjacent sequences: A178321 A178322 A178323 * A178325 A178326 A178327

Keyword

nonn

Author

Paul D. Hanna, Dec 21 2010

Status

approved