A392716 - OEIS

%I #20 Feb 16 2026 22:24:12

%S 1,0,6,36,510,8100,158046,3679956,98048670,2968618500,100316585886,

%T 3746574069876,153219398980830,6809706158264100,326828820016946526,

%U 16846193687562635796,928140176998059087390,54431376733124465919300,3385445099948855523111966

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

%H Vincenzo Librandi, <a href="/A392716/b392716.txt">Table of n, a(n) for n = 1..300</a>

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

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

%F 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

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x-(exp(x)-1)^3)))

%o (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

%Y Cf. A391954.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Jan 20 2026