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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Cf. A024786, A138785.

Sequence in context: A030746 A315848 A321450 * A005302 A028324 A074342

Adjacent sequences:  A301310 A301311 A301312 * A301314 A301315 A301316

KEYWORD

nonn

AUTHOR

Emily Anible, Apr 03 2018

EXTENSIONS

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

STATUS

approved

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Last modified May 16 05:56 EDT 2022. Contains 353693 sequences. (Running on oeis4.)