A302348 a(n) = Sum_{p in P} (H(2,p)^2)/2, where P is the set of partitions of n, and H(2,p) is the number of hooks of length 2 in p.

0, 0, 1, 1, 4, 5, 14, 18, 37, 50, 90, 122, 199, 270, 415, 559, 820, 1096, 1556, 2060, 2847, 3736, 5057, 6576, 8747, 11279, 14788, 18916, 24493, 31097, 39838, 50225, 63737, 79833, 100471, 125076, 156237, 193394, 239956, 295443, 364334, 446349, 547360, 667440

Offset: 0

Author

Emily Anible, Apr 05 2018

Keywords

• nonn

Comments

This sequence is part of the contribution to the b^2 term of the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two.

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

Examples

			For a(6), we sum over partitions of six. For each partition, we count 1 for each hook of length 2, then square the total in each partition. We divide the final result in half to get a(6).
6............1^2 = 1
5,1..........1^2 = 1
4,2..........2^2 = 4
4,1,1........2^2 = 4
3,3..........2^2 = 4
3,2,1........0^2 = 0
3,1,1,1......2^2 = 4
2,2,2........2^2 = 4
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.............28/2=14
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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.

Crossrefs

Cf. A024786, A302347.

Programs

Maple

b:= proc(n, i, p, l) option remember; `if`(n=0, p^2,
      `if`(i>n, 0, b(n, i+1, p, 1)+add(b(n-i*j, i+1, p+
      `if`(j>1, 1, 0)+l, 0), j=1..n/i)))
    end:
a:= n-> b(n, 1, 0$2)/2:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 06 2018

Mathematica

b[n_, i_, p_, l_] := b[n, i, p, l] = If[n == 0, p^2, If[i > n, 0, b[n, i + 1, p, 1] + Sum[b[n - i*j, i+1, p + If[j > 1, 1, 0]+l, 0], {j, 1, n/i}]]];
a[n_] := b[n, 1, 0, 0]/2;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2018, after Alois P. Heinz *)

Formula

G.f: (q^2*(1+q^2+2*q^4))/((1-q^2)*(1-q^4)*Product_{i>0} (1-q^i)).