A380638 Expansion of e.g.f. exp(x*G(4*x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 33, 2209, 226753, 31555521, 5557183201, 1185423664993, 297171500140929, 85638231765516673, 27896677183469054881, 10137203757416219332641, 4065668625283435566910273, 1783936343221839549449049409, 850091650335726912762794748513, 437197222292805469886634467693281

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

E.g.f.: exp( (G(4*x)-1)/4 ), where G(x) is described above.

a(n) = (n-1)! * Sum_{k=0..n-1} 4^k * binomial(4*n,k)/(n-k-1)! for n > 0.

a(n+1) = 4^n * n! * LaguerreL(n, 3*n+4, -1/4).

a(n) ~ 2^(10*n - 1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 1/12)). - Vaclav Kotesovec, Jan 29 2025

a(n) = (-4)^(n-1)*U(1-n, 2+3*n, -1/4), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025

Prog

(PARI) a(n) = if(n==0, 1, 4^(n-1)*(n-1)!*pollaguerre(n-1, 3*n+1, -1/4));

Crossrefs

Cf. A002293, A380516.

Sequence in context: A267672 A283533 A294773 * A294611 A294954 A372903

Adjacent sequences: A380635 A380636 A380637 * A380639 A380640 A380641

Keyword

nonn

Author

Seiichi Manyama, Jan 28 2025

Status

approved