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Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).
3

%I #12 Dec 15 2023 08:00:36

%S 0,0,0,0,1,5,15,35,105,756,6510,46530,289245,1892605,16187171,

%T 170721915,1833783770,18875258780,196470797580,2255939795436,

%U 29179692064545,401813199660285,5612352516200815,79620308330422475,1182881543312932386

%N Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

%F a(0) = 0; a(n) = binomial(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,4) * k * a(k).

%F a(n) ~ (n-1)! / (4*LambertW(3^(1/4)/2^(5/4)))^n. - _Vaclav Kotesovec_, Aug 08 2021

%F a(n) = n! * Sum_{k=1..floor(n/4)} k^(n-4*k-1)/(24^k * (n-4*k)!). - _Seiichi Manyama_, Dec 14 2023

%t nmax = 24; CoefficientList[Series[-Log[1 - x^4 Exp[x]/4!], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 0; a[n_] := a[n] = Binomial[n, 4] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

%Y Cf. A000332, A009444, A145454, A292955, A305990, A346752, A346753, A346754.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Aug 01 2021