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

0,5

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.

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.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A024786, A302347.

Sequence in context: A079362 A222364 A222372 * A029862 A092432 A041669

Adjacent sequences: A302345 A302346 A302347 * A302349 A302350 A302351

Keyword

nonn

Author

Emily Anible, Apr 05 2018

Status

approved