A392716 Expansion of e.g.f. Series_Reversion(x - (exp(x) - 1)^3).

1, 0, 6, 36, 510, 8100, 158046, 3679956, 98048670, 2968618500, 100316585886, 3746574069876, 153219398980830, 6809706158264100, 326828820016946526, 16846193687562635796, 928140176998059087390, 54431376733124465919300, 3385445099948855523111966

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (3*k)!/k! * Stirling2(n-1+k,3*k).

E.g.f. A(x) satisfies A(x) = x + (exp(A(x)) - 1)^3.

a(n) ~ sqrt(sqrt(5) + 2/phi^(2/3) - 2*phi^(2/3)) * 2^(n - 1/2) * 3^(n-1) * n^(n-1) / (5^(1/4) * exp(n) * (2/phi^(1/3) - 2*phi^(1/3) + 6*log((2 + 1/phi^(4/3) + phi^(4/3))/3))^(n - 1/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Feb 01 2026

Mathematica

Table[Sum[(3*k)!/k!*StirlingS2[n-1+k, 3*k], {k, 0, (n-1)/2}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 01 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x-(exp(x)-1)^3)))
(Magma) [&+[Factorial(3*k)/Factorial(k)*StirlingSecond(n-1+k, 3*k): k in [0..Floor((n-1)/2)]]: n in [1..25]]; // Vincenzo Librandi, Feb 14 2026

Crossrefs

Cf. A391954.

Sequence in context: A366654 A396096 A367488 * A339300 A392720 A296389

Adjacent sequences: A392711 A392712 A392713 * A392717 A392718 A392719

Keyword

nonn

Author

Seiichi Manyama, Jan 20 2026

Status

approved