

A234937


Triangle read by rows of coefficients of polynomials generated by the Han/NekrasovOkounkov formula.


4



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
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OFFSET

0,4


COMMENTS

Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1q^k)^(b1) = 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 GuoNiu Han's reference below gives the hooklength formula:
p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1b/(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], 20112012; Acta Arith. 160 (2013), 303315.


FORMULA

E.g.f.: Product_{k>0} (1q^k)^(b1).
Recurrence: With p_0(b) = 1, p_n(b) = (n1)!(b1) Sum_{m=1..n} (sigma(m) p_{nm}(b)) / ((nm)!}=) , sigma being the divisor function.


EXAMPLE

The coefficient of q^3 in the indeterminate power is (1/6) (1829b+12b^2b^3).


MATHEMATICA

nn=10;
Clear[b]; PolyTable = Table[0, {n, 1, nn}];
PolyTable[[1]]=1b;
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: nexttolast entry of row n, (1)^(n1) * n(3n1)/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.
Sequence in context: A082051 A196848 A266699 * A210590 A108446 A283263
Adjacent sequences: A234934 A234935 A234936 * A234938 A234939 A234940


KEYWORD

tabl,sign,easy


AUTHOR

William J. Keith, Jan 01 2014


STATUS

approved



