OFFSET
0,7
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.
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.
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emily Anible, Apr 03 2018
EXTENSIONS
a(10)-a(44) from Alois P. Heinz, Apr 03 2018
STATUS
approved