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A380636 Expansion of e.g.f. exp(x*C(2*x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

%I A380636 #16 Mar 16 2025 10:08:55
%S A380636 1,1,9,145,3409,105921,4102681,190630609,10340890785,641787925249,
%T A380636 44866443580201,3489524955627921,298914951848510449,
%U A380636 27966383049400396225,2837759948683874979129,310425081738609550495441,36418950255827044479693121,4561668082989623411575958529
%N A380636 Expansion of e.g.f. exp(x*C(2*x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.
%H A380636 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A380636 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A380636 E.g.f.: exp( (C(2*x)-1)/2 ), where C(x) is described above.
%F A380636 a(n) = (n-1)! * Sum_{k=0..n-1} 2^k * binomial(2*n,k)/(n-k-1)! for n > 0.
%F A380636 a(n+1) = 2^n * n! * LaguerreL(n, n+2, -1/2).
%F A380636 From _Vaclav Kotesovec_, Jan 29 2025: (Start)
%F A380636 E.g.f.: exp((1 - sqrt(1 - 8*x))^2 / (16*x)).
%F A380636 a(n) ~ 2^(3*n - 1/2) * n^(n-1) / exp(n - 1/2). (End)
%F A380636 a(n) = (-2)^(n-1)*U(1-n, 2+n, -1/2), where U is the Tricomi confluent hypergeometric function. - _Stefano Spezia_, Jan 29 2025
%F A380636 E.g.f.: exp( Series_Reversion( x/(1+2*x)^2 ) ). - _Seiichi Manyama_, Mar 16 2025
%t A380636 nmax = 20; CoefficientList[Series[E^((1 - Sqrt[1 - 8*x])^2 / (16*x)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jan 29 2025 *)
%o A380636 (PARI) a(n) = if(n==0, 1, 2^(n-1)*(n-1)!*pollaguerre(n-1, n+1, -1/2));
%Y A380636 Cf. A000108, A251568.
%K A380636 nonn
%O A380636 0,3
%A A380636 _Seiichi Manyama_, Jan 28 2025