A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

1, 7, 33, 130, 463, 1557, 5031, 15877, 49240, 151116, 460173, 1394645, 4212071, 12693724, 38195286, 114817389, 344911117, 1035659955, 3108817911, 9330152740, 27997803871, 84008165515, 252053831034, 756220333901, 2268778132337, 6806569134920, 20420175154486

Offset: 3

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A256130.

Cf. A320545.

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,3):
seq(a(n), n=3..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, 3];
Table[a[n], {n, 3, 35}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

Formula

a(n) ~ c * 3^n, where c = 1/(6*Product_{n>=1} (1-1/3^n)) = 1/(6*QPochhammer[1/3, 1/3]) = 1/(6*A100220) = 0.297552056999755698394581... . - Vaclav Kotesovec, Jun 01 2015