A373520 Expansion of e.g.f. exp(x/(1 - x^4)^(1/2)).
1, 1, 1, 1, 1, 61, 361, 1261, 3361, 143641, 1829521, 12501721, 59922721, 2173048021, 44315751481, 478799701381, 3492321094081, 116722067432881, 3290135175240481, 50242015215929521, 508061488330088641, 16418736123292904941, 585427887134915295241
Offset: 0
Programs
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Mathematica
nmax = 25; CoefficientList[Series[E^(x/(1 - x^4)^(1/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *) -
PARI
a(n) = n!*sum(k=0, n\4, binomial(n/2-k-1, k)/(n-4*k)!);
Formula
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/2-k-1,k)/(n-4*k)!.
a(n) == 1 mod 60.
From Vaclav Kotesovec, Sep 03 2025: (Start)
Recurrence: (n-8)*a(n) = (n-8)*a(n-2) + 3*(n-4)*(n-3)*(n-2)*(n^2 - 11*n + 20)*a(n-4) + 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*a(n-6) - 3*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n^2 - 13*n + 32)*a(n-8) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-10) + (n-12)*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-12).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3).
(End)