A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

1, 1, -1, 4, -5, 1, 18, -29, 12, -1, 120, -218, 119, -22, 1, 840, -1814, 1285, -345, 35, -1, 7920, -18144, 14674, -5205, 805, -51, 1, 75600, -196356, 185080, -79219, 16450, -1624, 70, -1, 887040, -2427312, 2515036, -1258628, 324569, -43568, 2954, -92, 1

Offset: 0

Author

William J. Keith, Jan 01 2014

Keywords

• tabl
• sign
• easy

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.

Examples

			The coefficient of q^3 in the indeterminate power is (1/6) (18-29b+12b^2-b^3).
		

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.

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.

Programs

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]]

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.