A352624 - OEIS

%I #14 Mar 25 2022 10:13:53

%S 1,1,3,8,31,122,579,2886,16139,95358,611111,4128830,29709695,

%T 224400022,1785322699,14841968646,129015458195,1167021383902,

%U 10979895178511,107113768171950,1082508179141031,11308614423992102,121995294474174963,1356835055606851286,15542964081299602811

%N Expansion of e.g.f. exp(exp(x) + cosh(x) - 2).

%H Seiichi Manyama, <a href="/A352624/b352624.txt">Table of n, a(n) for n = 0..566</a>

%F a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 + (-1)^k) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A000110(n-2*k).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A000807(k) * A003724(n-2*k).

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       a(n-k)*binomial(n-1, k-1)*(2-(k mod 2)), k=1..n))

%p     end:

%p seq(a(n), n=0..24);  # _Alois P. Heinz_, Mar 24 2022

%t nmax = 24; CoefficientList[Series[Exp[Exp[x] + Cosh[x] - 2], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 + (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

%Y Cf. A000110, A000807, A001861, A003724, A005046, A352327, A352617.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 24 2022