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

1, 4, 12, 30, 72, 160, 351, 743, 1561, 3219, 6616, 13456, 27312, 55139, 111166, 223472, 448902, 900305, 1804838, 3615137, 7239325, 14490368, 29000050, 58025059, 116090823, 232234573, 464554483, 929220024, 1858618215, 3717468189, 7435305664, 14871092926

Offset: 2

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=2 of A256130.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,2):
seq(a(n), n=2..35);

Mathematica

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[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 2];
a /@ Range[2, 35] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

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