A358265 Expansion of e.g.f. 1/(1 - x * exp(x^3/6)).
1, 1, 2, 6, 28, 160, 1080, 8470, 76160, 771120, 8671600, 107245600, 1446984000, 21150929800, 332950217600, 5615507898000, 101024594070400, 1931055071545600, 39082823446867200, 834945681049480000, 18776164188349568000, 443348081412556320000
Offset: 0
Programs
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Maple
g := 1/(1-x*exp(x^3/6)) ; taylor(%,x=0,70) ; L := gfun[seriestolist](%) ; seq( op(i,L)*(i-1)!,i=1..nops(L)) ; # R. J. Mathar, Mar 13 2023
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^3/6)))) -
PARI
a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*k!));
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(6^k * k!).
a(n) ~ n! / ((1 + LambertW(1/2)) * (2*LambertW(1/2))^(n/3)). - Vaclav Kotesovec, Nov 13 2022