login
A356796
E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^3).
3
1, 0, 2, 3, 92, 450, 14454, 141540, 4980128, 78711696, 3048567480, 68677353360, 2930551701384, 86832573553440, 4079649847428960, 150444517302424800, 7768028697749806080, 342721736137376184960, 19392702029822685015360, 994397473912386435004800
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(3 * x * log(1-x))/3 ).
E.g.f.: A(x) = ( LambertW(3 * x * log(1-x))/(3 * x * log(1-x)) )^(1/3).
PROG
(PARI) a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*log(1-x))/3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*log(1-x))/(3*x*log(1-x)))^(1/3)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 28 2022
STATUS
approved