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.
1, 1, 9, 145, 3409, 105921, 4102681, 190630609, 10340890785, 641787925249, 44866443580201, 3489524955627921, 298914951848510449, 27966383049400396225, 2837759948683874979129, 310425081738609550495441, 36418950255827044479693121, 4561668082989623411575958529
Offset: 0
Keywords
Programs
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Mathematica
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 *) -
PARI
a(n) = if(n==0, 1, 2^(n-1)*(n-1)!*pollaguerre(n-1, n+1, -1/2));
Formula
E.g.f.: exp( (C(2*x)-1)/2 ), where C(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 2^k * binomial(2*n,k)/(n-k-1)! for n > 0.
a(n+1) = 2^n * n! * LaguerreL(n, n+2, -1/2).
From Vaclav Kotesovec, Jan 29 2025: (Start)
E.g.f.: exp((1 - sqrt(1 - 8*x))^2 / (16*x)).
a(n) ~ 2^(3*n - 1/2) * n^(n-1) / exp(n - 1/2). (End)
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
E.g.f.: exp( Series_Reversion( x/(1+2*x)^2 ) ). - Seiichi Manyama, Mar 16 2025