A350291 - OEIS

%I #14 Nov 18 2023 03:29:00

%S 1,2,18,320,9170,376992,20773368,1464890856,127857825810,

%T 13468041055040,1678544187282248,243601216117107576,

%U 40628915629002303512,7702195195122395812600,1644169152021779307173400,392035147808725683863079120,103680698454480621919621969170

%N a(n) = 4^n*lim_{x->0} F(2*n) where F(n) = (d/dx)F(n-1) for n >= 1 and F(0) = exp(BesselI(0, x) - 1).

%H Vaclav Kotesovec, <a href="/A350291/b350291.txt">Table of n, a(n) for n = 0..55</a>

%F The Bell formula for n > 0 is:

%F a(n) = Sum_{k=1..2n} Y_{2n,k}(d_i), where Y_{n,k} is the partial Bell polynomial with inputs given by d_{2i} = binomial(2i,i) and d_{2i+1} = 0. - _Geoff Goehle_, Mar 11 2022

%p F := proc(n) option remember;

%p ifelse(n = 0, exp(BesselI(0, x) - 1), simplify(diff(F(n-1), x))) end:

%p a := n -> 4^n*limit(F(2*n), x=0): seq(a(n), n = 0..16);

%t Table[Sum[BellY[n, k, Flatten[Table[{0, Binomial[j+1, (j+1)/2]}, {j, 1, n-k+1, 2}]]], {k, 0, n}], {n, 0, 32, 2}] (* _Geoff Goehle_, Mar 11 2022 *)

%Y Cf. A352284, row sums of A350462.

%K nonn

%O 0,2

%A _Peter Luschny_, Mar 11 2022