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

2,2

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.

Formula

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

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 *)

Crossrefs

Column k=2 of A256193.

Cf. A094533.

Sequence in context: A322618 A260057 A260150 * A376710 A007678 A339493

Adjacent sequences: A258469 A258470 A258471 * A258473 A258474 A258475

Keyword

nonn

Author

Alois P. Heinz, May 31 2015

Status

approved