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
KEYWORD
nonn
AUTHOR
Emily Anible, Apr 05 2018
STATUS
approved