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a(0) = 1; a(n+1) = a(n)*2n + 2n + 1.
0

%I #11 Mar 14 2023 19:16:35

%S 1,1,5,25,157,1265,12661,151945,2127245,34035937,612646885,

%T 12252937721,269564629885,6469551117265,168208329048917,

%U 4709833213369705,141294996401091181,4521439884834917825,153728956084387206085,5534242419037939419097,210301211923441697925725

%N a(0) = 1; a(n+1) = a(n)*2n + 2n + 1.

%C Sum of reciprocals = 2.2472460058071954531775930749...

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

%p `if`(n=0, 1, 2*((n-1)*a(n-1)+n)-1)

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 14 2023

%t nxt[{n_,a_}]:={n+1,a*2n+2n+1}; Transpose[NestList[nxt,{1,1},20]][[2]] (* _Harvey P. Dale_, Aug 06 2016 *)

%o (PARI) sum2x(n) = { s=1; sr=0; forstep(x=2,n,2, s=x*(s+1)+1; print1(s","); sr += 1.0/s; ); print(); print(sr) }

%Y a(n) = 2 * A010844(n-1) - 1 = A007566(n-1) - 2n - 2 for n>=1.

%K nonn,easy

%O 0,3

%A _Cino Hilliard_, Aug 19 2003

%E a(0)=1 prepended and edited by _Alois P. Heinz_, Mar 14 2023