A365527 - OEIS

%I #13 Sep 13 2023 02:09:54

%S 0,0,1,3,7,15,32,84,393,2901,23339,180565,1327404,9364732,64197317,

%T 433372411,2928720335,20264399483,147807954692,1170622475408,

%U 10229966924581,97922117830589,1001744359476291,10661002700183905,115706501336004984

%N a(n) = Sum_{k=0..floor((n-2)/4)} Stirling2(n,4*k+2).

%F Let A(0)=1, B(0)=0, C(0)=0 and D(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*D(k). A365525(n) = A(n), A365526(n) = B(n), a(n) = C(n) and A099948(n) = D(n).

%F G.f.: Sum_{k>=0} x^(4*k+2) / Product_{j=1..4*k+2} (1-j*x).

%t a[n_] := Sum[StirlingS2[n, 4*k+2], {k, 0, Floor[(n-2)/4]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 13 2023 *)

%o (PARI) a(n) = sum(k=0, (n-2)\4, stirling(n, 4*k+2, 2));

%Y Cf. A099948, A365525, A365526.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Sep 08 2023