A302347 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.
0, 0, 1, 1, 3, 4, 10, 13, 25, 34, 59, 80, 127, 172, 260, 349, 505, 673, 946, 1248, 1711, 2238, 3010, 3902, 5162, 6637, 8663, 11051, 14253, 18051, 23047, 28988, 36677, 45840, 57538, 71485, 89082, 110062, 136269, 167487, 206138, 252132, 308640, 375777, 457698
Offset: 0
Keywords
Examples
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. 6............0^2 = 0 5,1..........0^2 = 0 4,2..........0^2 = 0 4,1,1........1^2 = 1 3,3..........1^2 = 1 3,2,1........0^2 = 0 3,1,1,1......1^2 = 1 2,2,2........1^2 = 1 2,2,1,1......2^2 = 4 2,1,1,1,1....1^2 = 1 1,1,1,1,1,1..1^2 = 1 -------------------- Total.............10
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..2000
- Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398 [math.CO], 2008.
- Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.
- W. J. Keith, Restricted k-color partitions, Ramanujan Journal (2016) 40: 71.
Programs
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Maple
b:= proc(n, i, p) option remember; `if`(n=0 or i=1, ( `if`(n>1, 1, 0)+p)^2, add(b(n-i*j, i-1, `if`(j>1, 1, 0)+p), j=0..n/i)) end: a:= n-> b(n$2, 0): seq(a(n), n=0..60); # Alois P. Heinz, Apr 05 2018 -
Mathematica
Array[Total[Count[Split@ #, ?(Length@ # > 1 &)]^2 & /@ IntegerPartitions[#]] &, 44] (* _Michael De Vlieger, Apr 07 2018 *) b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, ( If[n > 1, 1, 0] + p)^2, Sum[b[n - i*j, i - 1, If[j > 1, 1, 0] + p], {j, 0, n/i}]]; a[n_] := b[n, n, 0]; a /@ Range[0, 60] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)
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Python
def sum_square_freqs_greater_one(freq_list): tot = 0 for f in freq_list: count = 0 for i in f: if i > 1: count += 1 tot += count*count return tot def frequencies(partition, n): tot = 0 freq_list = [] i = 0 for p in partition: freq = [0 for i in range(n+1)] for i in p: freq[i] += 1 for f in freq: if f == 0: tot += 1 freq_list.append(freq) return freq_list
Formula
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.
G.f: (q^2*(1+q^4))/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 22 2018
Comments