A258472 Number of partitions of n into two sorts of parts having exactly 2 parts of the second sort.

1, 4, 11, 24, 49, 89, 158, 262, 428, 667, 1033, 1542, 2289, 3313, 4765, 6717, 9427, 13011, 17882, 24260, 32763, 43775, 58268, 76837, 100953, 131629, 171003, 220683, 283877, 363016, 462794, 587005, 742332, 934536, 1173293, 1467022, 1829538, 2273365, 2817858

Offset: 2

Author

Alois P. Heinz, May 31 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000
• Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

Crossrefs

Column k=2 of A256193.

Cf. A094533.

Programs

Maple

b:= proc(n, i) option remember; series(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
       binomial(j, t), t=0..min(2, j)), j=0..n/i))), x, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..40);

Mathematica

((Log[1 - x]^2 - Log[1 - x] Log[x] + QPolyGamma[1, x] (2 Log[1 - x] - Log[x] + QPolyGamma[1, x]) + QPolyGamma[1, 1, x])/(2 QPochhammer[x] Log[x]^2) + O[x]^45)[[3]] // Simplify (* Vladimir Reshetnikov, Nov 21 2016 *)
Table[SeriesCoefficient[1/QPochhammer[q + x, q], {x, 0, 2}, {q, 0, n}], {n, 0, 40}] // Simplify (* Vladimir Reshetnikov, Nov 22 2016 *)

Formula

a(n) = A094533(n)/2. - Vladimir Reshetnikov, Nov 21 2016