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

%I #20 Sep 27 2024 10:38:16

%S 1,4,11,24,49,89,158,262,428,667,1033,1542,2289,3313,4765,6717,9427,

%T 13011,17882,24260,32763,43775,58268,76837,100953,131629,171003,

%U 220683,283877,363016,462794,587005,742332,934536,1173293,1467022,1829538,2273365,2817858

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

%H Alois P. Heinz, <a href="/A258472/b258472.txt">Table of n, a(n) for n = 2..10000</a>

%H Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, <a href="https://arxiv.org/abs/2409.11268">Elementary symmetric partitions</a>, arXiv:2409.11268 [math.CO], 2024. See p. 20.

%F a(n) = A094533(n)/2. - _Vladimir Reshetnikov_, Nov 21 2016

%p b:= proc(n, i) option remember; series(`if`(n=0, 1,

%p `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*

%p binomial(j, t), t=0..min(2, j)), j=0..n/i))), x, 3)

%p end:

%p a:= n-> coeff(b(n$2), x, 2):

%p seq(a(n), n=2..40);

%t ((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 *)

%t Table[SeriesCoefficient[1/QPochhammer[q + x, q], {x, 0, 2}, {q, 0, n}], {n, 0, 40}] // Simplify (* _Vladimir Reshetnikov_, Nov 22 2016 *)

%Y Column k=2 of A256193.

%Y Cf. A094533.

%K nonn

%O 2,2

%A _Alois P. Heinz_, May 31 2015