A234937 - OEIS

%I #41 Nov 07 2020 14:20:39

%S 1,1,-1,4,-5,1,18,-29,12,-1,120,-218,119,-22,1,840,-1814,1285,-345,35,

%T -1,7920,-18144,14674,-5205,805,-51,1,75600,-196356,185080,-79219,

%U 16450,-1624,70,-1,887040,-2427312,2515036,-1258628,324569,-43568,2954,-92,1

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

%C Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1-q^k)^(b-1) = Sum n! p_n(b) q^n.

%C Each row is length 1+n, starting from n=0, and consists of the coefficients of one of the p_n(b).

%C 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:

%C p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1-b/(h_{ij}^2)).

%C 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.

%C Lehmer's conjecture on the tau function states that the evaluation at b=25 (A000594) is never 0.

%C The general diagonal and column are probably of combinatorial interest.

%H Seiichi Manyama, <a href="/A234937/b234937.txt">Rows n = 0..100, flattened</a>

%H G.-N. Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO], 2008.

%H W. J. Keith, <a href="http://arxiv.org/abs/1109.1236">Polynomial analogues of Ramanujan congruences for Han's hooklength formula</a>, arXiv:1109.1236 [math.CO], 2011-2012; Acta Arith. 160 (2013), 303-315.

%F E.g.f.: Product_{k>0} (1-q^k)^(b-1).

%F 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.

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

%t nn=10;

%t Clear[b]; PolyTable = Table[0, {n, 1, nn}];

%t PolyTable[[1]]=1-b;

%t For[n = 2, n <= nn, n++,

%t PolyTable[[n]] = Simplify[(((n - 1)!)*(b - 1))*(Sum[

%t        PolyTable[[n - m]]*(-1*DivisorSigma[1, m]/((n - m)!)), {m, 1,

%t         n - 1}] + (-1*DivisorSigma[1, n]))]];

%t LongTable = Table[Table[

%t    Which[k == 0, PartitionsP[n]*n!, k > 0,

%t     Coefficient[Expand[PolyTable[[n]]], b^k]], {k, 0, n}], {n, 1, nn}];

%t Flatten[PrependTo[LongTable,1]]

%Y Row entries sum to 0.

%Y A210590 is the unsigned version.

%Y Starting from row 0: final entry of row n, (-1)^n (A033999).

%Y From row 1: next-to-last entry of row n, (-1)^(n-1) * n(3n-1)/2 (signed version of A000326).

%Y First entry of row n, n! * p(n) (A053529).

%Y Second entry of row n, -1 * n! * (sum of reciprocals of all parts in partitions of n) (negatives of A057623).

%Y (Sum of absolute values of row entries)/n!: A000712.

%Y 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:

%Y b=0: A000041, the partition numbers,

%Y b=2: A010815, from Euler's Pentagonal Number Theorem,

%Y b=-1: A000712, partitions into 2 colors,

%Y b=-11: A005758, reciprocal of the square root of the tau function,

%Y b=-23: A006922, reciprocal of the tau function,

%Y b=13: A000735, square root of the tau function,

%Y b=25: A000594, Ramanujan's tau function.

%K tabl,sign,easy

%O 0,4

%A _William J. Keith_, Jan 01 2014