A168554 G.f.: A(x) = Sum_{n>=0} 2^(n^2)*A000108(n)*(1-2^n*x)^n*x^n where A000108 is the Catalan numbers.

1, 2, 28, 2304, 856576, 1351057408, 8846893121536, 238036693238677504, 26163011929227894194176, 11701653843176682031379644416, 21237338088859808279441141143699456

Offset

0,2

Comments

Compare the g.f. to: Sum_{n>=0} A000108(n)*(1-x)^n*x^n = 1/(1-x).

Links

Robert Israel, Table of n, a(n) for n = 0..56

Formula

a(n) = [x^n] 2/(1 + sqrt(1 - 4*2^n*(x-x^2))).

a(n) = Sum_{k=0..[n/2]} (-1)^k*2^(n(n-k))*C(n-k,k)*A000108(n-k).

Example

G.f.: A(x) = 1 + 2*x + 28*x^2 + 2304*x^3 + 856576*x^4 +...

Maple

S:= add(2^(n^2)*binomial(2*n, n)/(n+1)*(1-2^n*x)^n*x^n, n=0..30):
seq(coeff(S, x, n), n=0..30); # Robert Israel, Nov 13 2016

Mathematica

Table[Sum[(-1)^k*2^(n(n-k))*Binomial[n-k, k]*Binomial[2*(n-k), (n-k)]/(n-k+1), {k, 0, Floor[n/2]}], {n, 0, 20}] (* G. C. Greubel, Nov 13 2016 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, 2^(m^2)*binomial(2*m, m)/(m+1)*x^m*(1-2^m*x)^m)+x*O(x^n), n)}
(PARI) {a(n)=polcoeff(2/(1+sqrt(1-4*2^n*(x-x^2) +x*O(x^n))), n)}
(PARI) {a(n)=sum(k=0, n\2, (-1)^k*2^(n*(n-k))*binomial(n-k, k)*binomial(2*n-2*k, n-k)/(n-k+1))}

Crossrefs

Cf. A000108.

Sequence in context: A326366 A177400 A230700 * A152792 A085602 A058502

Adjacent sequences: A168551 A168552 A168553 * A168555 A168556 A168557

Keyword

nonn

Author

Paul D. Hanna, Nov 29 2009

Status

approved