A380638 - OEIS

%I #13 Jan 29 2025 07:57:05

%S 1,1,33,2209,226753,31555521,5557183201,1185423664993,297171500140929,

%T 85638231765516673,27896677183469054881,10137203757416219332641,

%U 4065668625283435566910273,1783936343221839549449049409,850091650335726912762794748513,437197222292805469886634467693281

%N 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

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

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

%F a(n) ~ 2^(10*n - 1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 1/12)). - _Vaclav Kotesovec_, Jan 29 2025

%F 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

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

%Y Cf. A002293, A380516.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 28 2025