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A346889
Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).
7
1, 0, 0, 1, 4, 10, 40, 315, 2296, 15204, 117720, 1127445, 11531740, 120909646, 1370809804, 17111895255, 227853866800, 3182209445640, 47003318806896, 737325061500009, 12187616610231540, 210930852047426770, 3821604062633503300, 72479758506840597451
OFFSET
0,5
LINKS
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,3) * a(n-k).
a(n) ~ n! / ((1 + LambertW(2^(1/3)/3^(2/3))) * 3^(n+1) * LambertW(2^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(6^k * (n-3*k)!). - Seiichi Manyama, May 13 2022
MATHEMATICA
nmax = 23; CoefficientList[Series[1/(1 - x^3 Exp[x]/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
PROG
(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^3*exp(x)/3!))) \\ Michel Marcus, Aug 06 2021
(PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(6^k*(n-3*k)!)); \\ Seiichi Manyama, May 13 2022
CROSSREFS
Column k=3 of A351703.
Sequence in context: A370997 A355993 A220817 * A114918 A224488 A149213
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 06 2021
STATUS
approved