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

0,4

Links

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

Formula

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

Example

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.

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]

Crossrefs

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

Sequence in context: A000223 A366044 A031328 * A053159 A345891 A279912

Adjacent sequences: A255177 A255178 A255179 * A255181 A255182 A255183

Keyword

nonn

Author

Geoffrey Critzer, Mar 19 2015

Status

approved