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

0,5

Comments

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.

It is of interest to enumerate and determine specific characteristics of partitions of n, considering each partition individually.

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A024786, A024788, A302300.

Sequence in context: A031367 A073443 A257494 * A092119 A362649 A143372

Adjacent sequences: A302344 A302345 A302346 * A302348 A302349 A302350

Keyword

nonn

Author

Emily Anible, Apr 05 2018

Status

approved