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URL: https://oeis.org/A390648

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A390648
E.g.f. A(x) satisfies A(x) = 1 + x^3*exp(x)*A(x)^2.
2
1, 0, 0, 6, 24, 60, 1560, 20370, 161616, 2782584, 59271120, 919417950, 17568715560, 454110435636, 10647755840904, 256825487321130, 7563662622999840, 229518120666486000, 6958840286128877856, 234938349337715982774, 8496977765543847712440, 311786432158975669090860
OFFSET
0,4
LINKS
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k) * Catalan(k)/(n-3*k)!.
E.g.f.: 2/(1 + sqrt(1 - 4*x^3*exp(x))).
a(n) ~ sqrt(2*(1 + LambertW(1/(3*2^(2/3))))) * n^(n-1) / (3^(n - 1/2) * exp(n) * LambertW(1/(3*2^(2/3)))^n). - Vaclav Kotesovec, Nov 18 2025
MATHEMATICA
Join[{1}, Table[Factorial[n]*Sum[k^(n-3*k)*Binomial[2*k, k]/((k+1)*Factorial[n-3*k]), {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vincenzo Librandi, Dec 27 2025 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(2*k, k)/((k+1)*(n-3*k)!));
(Magma) [Factorial(n)*&+[k^(n-3*k)*Binomial(2*k, k)/((k+1)*Factorial((n-3*k))): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 27 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 13 2025
STATUS
approved