OFFSET
0,4
COMMENTS
Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1-q^k)^(b-1) = Sum n! p_n(b) q^n.
Each row is length 1+n, starting from n=0, and consists of the coefficients of one of the p_n(b).
A210590 is an unsigned version using the form preferred by Nekrasov and Okounkov. This is the form for which Guo-Niu Han's reference below gives the hooklength formula:
p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1-b/(h_{ij}^2)).
Coefficients reduced mod 5 are those of 2 times Pascal's triangle and an alternating sign. Other primes have slightly more complex reduction behavior. See second link.
Lehmer's conjecture on the tau function states that the evaluation at b=25 (A000594) is never 0.
The general diagonal and column are probably of combinatorial interest.
LINKS
Seiichi Manyama, Rows n = 0..100, flattened
G.-N. Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.
W. J. Keith, Polynomial analogues of Ramanujan congruences for Han's hooklength formula, arXiv:1109.1236 [math.CO], 2011-2012; Acta Arith. 160 (2013), 303-315.
FORMULA
E.g.f.: Product_{k>0} (1-q^k)^(b-1).
Recurrence: With p_0(b) = 1, p_n(b) = (n-1)!(b-1) Sum_{m=1..n} (-sigma(m) p_{n-m}(b)) / ((n-m)!}=) , sigma being the divisor function.
EXAMPLE
The coefficient of q^3 in the indeterminate power is (1/6) (18-29b+12b^2-b^3).
MATHEMATICA
nn=10;
Clear[b]; PolyTable = Table[0, {n, 1, nn}];
PolyTable[[1]]=1-b;
For[n = 2, n <= nn, n++,
PolyTable[[n]] = Simplify[(((n - 1)!)*(b - 1))*(Sum[
PolyTable[[n - m]]*(-1*DivisorSigma[1, m]/((n - m)!)), {m, 1,
n - 1}] + (-1*DivisorSigma[1, n]))]];
LongTable = Table[Table[
Which[k == 0, PartitionsP[n]*n!, k > 0,
Coefficient[Expand[PolyTable[[n]]], b^k]], {k, 0, n}], {n, 1, nn}];
Flatten[PrependTo[LongTable, 1]]
CROSSREFS
Row entries sum to 0.
A210590 is the unsigned version.
Starting from row 0: final entry of row n, (-1)^n (A033999).
From row 1: next-to-last entry of row n, (-1)^(n-1) * n(3n-1)/2 (signed version of A000326).
First entry of row n, n! * p(n) (A053529).
Second entry of row n, -1 * n! * (sum of reciprocals of all parts in partitions of n) (negatives of A057623).
(Sum of absolute values of row entries)/n!: A000712.
Evaluations at various powers of b, divided by n!, enumerate multipartitions or powers of the eta function. Some special cases that appear in the OEIS:
b=0: A000041, the partition numbers,
b=2: A010815, from Euler's Pentagonal Number Theorem,
b=-1: A000712, partitions into 2 colors,
b=-11: A005758, reciprocal of the square root of the tau function,
b=-23: A006922, reciprocal of the tau function,
b=13: A000735, square root of the tau function,
b=25: A000594, Ramanujan's tau function.
KEYWORD
AUTHOR
William J. Keith, Jan 01 2014
STATUS
approved