A346396 - OEIS

%I #12 May 27 2022 14:36:21

%S 0,1,9,62,390,2384,14680,93680,635824,4697664,38442112,351331584,

%T 3582715136,40476303360,501863078912,6767130867712,98464775493632,

%U 1536203429306368,25564684461735936,451816479967608832,8448863295040978944,166627401783086415872,3455980532191764676608

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

%H Seiichi Manyama, <a href="/A346396/b346396.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = n! * Sum_{k=0..n-1} 4^k / ((n-k) * k!).

%F a(n) ~ exp(4) * (n-1)!. - _Vaclav Kotesovec_, Aug 09 2021

%F a(0) = 0, a(1) = 1, a(n) = (n+3) * a(n-1) - 4 * (n-1) * a(n-2) + 4^(n-1). - _Seiichi Manyama_, May 27 2022

%t nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[4 x], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[4^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=0; v[2]=1; for(i=2, n, v[i+1]=(i+3)*v[i]-4*(i-1)*v[i-1]+4^(i-1)); v; \\ _Seiichi Manyama_, May 27 2022

%Y Cf. A002104, A053487, A346394, A346395.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 15 2021