A346751 - OEIS

%I #15 Dec 15 2023 09:09:57

%S 0,0,0,1,4,10,10,-105,-1064,-6076,-16680,129525,2642860,25431406,

%T 130210444,-639438345,-26431524560,-382074099000,-3083015556624,

%U 5641134587049,726952330301940,14940678486798610,173111303303845060,258953439321230731,-43858702741534022936

%N Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

%H Seiichi Manyama, <a href="/A346751/b346751.txt">Table of n, a(n) for n = 0..471</a>

%F a(0) = 0; a(n) = binomial(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).

%F a(n) = n! * Sum_{k=1..floor(n/3)} (-1)^(k-1) * k^(n-3*k-1)/(6^k * (n-3*k)!). - _Seiichi Manyama_, Dec 14 2023

%t nmax = 24; CoefficientList[Series[Log[1 + x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 0; a[n_] := a[n] = Binomial[n, 3] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

%Y Cf. A000292, A009306, A145453, A346750, A346752, A346754.

%K sign

%O 0,5

%A _Ilya Gutkovskiy_, Aug 01 2021