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A258457 Number of partitions of n into parts of exactly 2 sorts which are introduced in ascending order.

%I A258457 #11 Dec 11 2020 04:17:28
%S A258457 1,4,12,30,72,160,351,743,1561,3219,6616,13456,27312,55139,111166,
%T A258457 223472,448902,900305,1804838,3615137,7239325,14490368,29000050,
%U A258457 58025059,116090823,232234573,464554483,929220024,1858618215,3717468189,7435305664,14871092926
%N A258457 Number of partitions of n into parts of exactly 2 sorts which are introduced in ascending order.
%H A258457 Alois P. Heinz, <a href="/A258457/b258457.txt">Table of n, a(n) for n = 2..1000</a>
%F A258457 a(n) ~ c * 2^n, where c = 1/Product_{n>=2} (1-1/2^n) = 1/(2*A048651) = 1.7313733097275318... . - _Vaclav Kotesovec_, Jun 01 2015
%p A258457 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258457       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258457     end:
%p A258457 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258457 a:= n-> T(n,2):
%p A258457 seq(a(n), n=2..35);
%t A258457 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
%t A258457 T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
%t A258457 a[n_] := T[n, 2];
%t A258457 a /@ Range[2, 35] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A258457 Column k=2 of A256130.
%K A258457 nonn
%O A258457 2,2
%A A258457 _Alois P. Heinz_, May 30 2015