A346750 - OEIS

%I #15 Dec 15 2023 08:27:20

%S 0,0,1,3,3,-20,-135,-189,3598,33300,39105,-2164085,-23831214,-5268042,

%T 3038813869,36984819795,-59749871880,-8207734934984,-105142191601887,

%U 482549202944307,37754304692254030,489494512692093090,-4466445363328684659,-271973408844483808517

%N Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

%H Seiichi Manyama, <a href="/A346750/b346750.txt">Table of n, a(n) for n = 0..457</a>

%F a(0) = 0; a(n) = binomial(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).

%F a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * k^(n-2*k-1)/(2^k * (n-2*k)!). - _Seiichi Manyama_, Dec 14 2023

%t nmax = 23; CoefficientList[Series[Log[1 + x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 0; a[n_] := a[n] = Binomial[n, 2] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

%Y Cf. A000217, A009306, A133189, A300455, A346751, A346752, A346753.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Aug 01 2021