A301313 a(n) = Sum_{p in P} binomial(H(2,p),2), where P is the set of partitions of n, and H(2,p) = number of hooks of size 2 in p.

0, 0, 0, 0, 1, 1, 6, 7, 18, 24, 49, 66, 116, 158, 255, 346, 525, 707, 1030, 1374, 1936, 2560, 3519, 4608, 6207, 8056, 10673, 13735, 17942, 22906, 29569, 37469, 47864, 60235, 76249, 95335, 119705, 148770, 185447, 229182, 283810, 348903, 429498, 525411, 643244

Offset: 0

Author

Emily Anible, Apr 03 2018

Keywords

• nonn

Comments

This sequence is part of the contribution to the quadratic b^2 term of a 2-truncation of the Han/Nekrasov-Okounkov hooklength formula (2-truncation here being the limiting of hook sizes counted by the formula to only those of size 1 or 2). Exploring this sequence may lead to more general formulas regarding the hooklength formula for larger hooks, or the entire contribution to the quadratic term of the formula.

Examples

			For n=6, we sum over the partitions of 6. For each partition, we calculate binomial(number of hooks of size 2 in partition, 2):
6............binomial(1,2) = 0
5,1..........binomial(1,2) = 0
4,2..........binomial(2,2) = 1
4,1,1........binomial(2,2) = 1
3,3..........binomial(2,2) = 1
3,2,1........binomial(0,2) = 0
3,1,1,1......binomial(2,2) = 1
2,2,2........binomial(2,2) = 1
2,2,1,1......binomial(2,2) = 1
2,1,1,1,1....binomial(1,2) = 0
1,1,1,1,1,1..binomial(1,2) = 0
------------------------------
Total........................6
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1900 from Alois P. Heinz)
• 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.

Crossrefs

Cf. A024786, A138785.

Programs

Maple

b:= proc(n, i, p, l) option remember; `if`(n=0, p*(p-1)/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):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 05 2018

Mathematica

b[n_, i_, p_, l_] := b[n, i, p, l] = If[n == 0, p*(p-1)/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];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)
Table[Sum[(2*k - 5 - (-1)^(k/2))*(1 + (-1)^k)/4 * PartitionsP[n-k], {k, 1, n}], {n, 0, 60}] (* Vaclav Kotesovec, Oct 06 2018 *)

Formula

G.f.: (q^4+3*q^6)/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j). - Emily Anible, May 18 2018

a(n) ~ sqrt(3) * exp(Pi*sqrt((2*n)/3)) / (4*Pi^2). - Vaclav Kotesovec, Oct 06 2018

Extensions

a(10)-a(44) from Alois P. Heinz, Apr 03 2018