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%I #32 May 29 2024 12:25:29
%S 0,0,1,1,3,4,10,13,25,34,59,80,127,172,260,349,505,673,946,1248,1711,
%T 2238,3010,3902,5162,6637,8663,11051,14253,18051,23047,28988,36677,
%U 45840,57538,71485,89082,110062,136269,167487,206138,252132,308640,375777,457698
%N a(n) = Sum of (Y(2,p)^2) over the partitions p of n, Y(2,p) = number of part sizes with multiplicity 2 or greater in p.
%C This sequence is part of the contribution to the b^2 term of C_{1-b,2}(q) for(1-b,2)-colored partitions - partitions in which we can label parts any of an indeterminate 1-b colors, but are restricted to using only 2 of the colors per part size. This formula is known to match the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two up to the linear term in b.
%C It is of interest to enumerate and determine specific characteristics of partitions of n, considering each partition individually.
%H Alois P. Heinz, <a href="/A302347/b302347.txt">Table of n, a(n) for n = 0..2000</a>
%H Guo-Niu Han, <a href="https://arxiv.org/abs/0805.1398">The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications</a>, arXiv:0805.1398 [math.CO], 2008.
%H Guo-Niu Han, <a href="https://doi.org/10.5802/aif.2515">The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications</a>, Annales de l'institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.
%H W. J. Keith, <a href="https://doi.org/10.1007/s11139-015-9704-x">Restricted k-color partitions</a>, Ramanujan Journal (2016) 40: 71.
%F a(n) = Sum_{p in P(n)} (H(2,p)^2 + 2*A024786 - 2*A024788), where P(n) is the set of partitions of n, and H(2,p) is the hooks of length 2 in partition p.
%F G.f: (q^2*(1+q^4))/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).
%F a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - _Vaclav Kotesovec_, May 22 2018
%e For a(6), we sum over partitions of six. For each partition, we count 1 for each part which appears more than once, then square the total in each partition.
%e 6............0^2 = 0
%e 5,1..........0^2 = 0
%e 4,2..........0^2 = 0
%e 4,1,1........1^2 = 1
%e 3,3..........1^2 = 1
%e 3,2,1........0^2 = 0
%e 3,1,1,1......1^2 = 1
%e 2,2,2........1^2 = 1
%e 2,2,1,1......2^2 = 4
%e 2,1,1,1,1....1^2 = 1
%e 1,1,1,1,1,1..1^2 = 1
%e --------------------
%e Total.............10
%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (
%p `if`(n>1, 1, 0)+p)^2, add(b(n-i*j, i-1,
%p `if`(j>1, 1, 0)+p), j=0..n/i))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Apr 05 2018
%t Array[Total[Count[Split@ #, _?(Length@ # > 1 &)]^2 & /@ IntegerPartitions[#]] &, 44] (* _Michael De Vlieger_, Apr 07 2018 *)
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (
%t If[n > 1, 1, 0] + p)^2, Sum[b[n - i*j, i - 1,
%t If[j > 1, 1, 0] + p], {j, 0, n/i}]];
%t a[n_] := b[n, n, 0];
%t a /@ Range[0, 60] (* _Jean-François Alcover_, Jun 06 2021, after _Alois P. Heinz_ *)
%o (Python)
%o def sum_square_freqs_greater_one(freq_list):
%o tot = 0
%o for f in freq_list:
%o count = 0
%o for i in f:
%o if i > 1:
%o count += 1
%o tot += count*count
%o return tot
%o def frequencies(partition, n):
%o tot = 0
%o freq_list = []
%o i = 0
%o for p in partition:
%o freq = [0 for i in range(n+1)]
%o for i in p:
%o freq[i] += 1
%o for f in freq:
%o if f == 0:
%o tot += 1
%o freq_list.append(freq)
%o return freq_list
%Y Cf. A024786, A024788, A302300.
%K nonn
%O 0,5
%A _Emily Anible_, Apr 05 2018
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