A255180 Number of partitions of n in which two summands (of each size) are designated.

1, 0, 1, 3, 7, 10, 20, 24, 45, 61, 103, 140, 246, 325, 517, 728, 1086, 1472, 2184, 2918, 4197, 5638, 7875, 10497, 14625, 19272, 26354, 34804, 46992, 61490, 82471, 107163, 142128, 184141, 241701, 311282, 406164, 519755, 672726, 858110, 1102872

Offset: 0

Author

Geoffrey Critzer, Mar 19 2015

Keywords

• nonn

Examples

			a(4)=7. In order to designate two summands of each size, the multiplicity of each summand must be at least two. For n=4 we consider only the partitions 2+2 and 1+1+1+1.  In the first case there is 1 way and in the second case there are 6 ways.  1 + 6 = 7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A077285, A070933 (where any number of summands of any size are designated).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*binomial(j, 2), j=2..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2015

Mathematica

nn = 40; CoefficientList[Series[Product[1 + x^(2 n)/(1 - x^n)^3, {n, 1, nn}], {x, 0, nn}], x]

Formula

G.f.: Product_{n>=1} 1 + x^(2*n)/(1 - x^n)^3.