A188456 G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1 - 2^n*x)^(n+1).

1, 1, 4, 44, 1216, 80640, 12460032, 4393091072, 3479212916736, 6113821454237696, 23602899265140031488, 198562423940692641316864, 3615246879908004653107773440, 141631725381846630255125115961344

Offset

0,3

Comments

G.f. satisfies a variant of an identity of the Catalan numbers (A000108):

1 = Sum_{n>=0} A000108(n)*x^n*(1 - x)^(n+1).

Links

Table of n, a(n) for n=0..13.

Formula

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

Example

G.f.: 1 = (1-x) + x*(1-2*x)^2 + 4*x^2*(1-4*x)^3 + 44*x^3*(1-8*x)^4 + 1216*x^4*(1-16*x)^5 + 80640*x^5*(1-32*x)^6 + ...

Mathematica

a[0] = 1; a[n_] := a[n] = SeriesCoefficient[1-Sum[a[k]*x^k*(1-2^k*x)^(k+1), {k, 0, n-1}], {x, 0, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 09 2017 *)

Prog

(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-2^k*x+x*O(x^n))^(k+1)), n)}

Crossrefs

Cf. A188455, A188457, A188458.

Sequence in context: A137783 A136552 A155556 * A306372 A217473 A127635

Adjacent sequences: A188453 A188454 A188455 * A188457 A188458 A188459

Keyword

nonn

Author

Paul D. Hanna, Mar 31 2011

Status

approved