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Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)_n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product_{i=0..n-1} (1-x q^i).
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%I #33 May 19 2021 01:33:51

%S 1,1,1,0,1,1,0,1,2,1,0,1,1,0,1,2,4,2,4,2,4,2,1,0,1,1,0,1,2,4,6,7,8,12,

%T 12,14,12,12,8,7,6,4,2,1,0,1,1,0,1,2,4,6,12,12,21,26,37,40,55,52,61,

%U 60,61,52,55,40,37,26,21,12,12,6,4,2,1,0,1,1,0,1,2,4,6,12,18,26,38,57,76

%N Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)_n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product_{i=0..n-1} (1-x q^i).

%C f_n has degree n(n-1), so n-th row of table has n(n-1)+1 entries. Each row is palindromic. The sum of the terms in the n-th row is n!. The first n+1 terms of the n-th row are the same as the first n terms of A052847.

%C The 'major index' maj(p) of a permutation p = a_1 a_2 ... a_n is the sum of all i such that a_i > a_(i+1). f_n(q) = Sum_p q^(maj(p)+maj(p^(-1))), where the sum is over all permutations of {1,2,...,n}.

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.

%H Alois P. Heinz, <a href="/A081285/b081285.txt">Rows n = 0..40, flattened</a>

%H Zhipeng Lu, <a href="https://arxiv.org/abs/2103.02168">Symmetric permutation invariants in some tensor products</a>, arXiv:2103.02168 [math.CO], 2021.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol.</a>

%F f_n(q) = Sum_{r=1..n} (-1)^(r+1) q^(r(r-1)/2) (q)_(n-1) (q)_n / ((q)_(r) ((q)_(n-r))^2) f_(n-r)(q) for n>=1.

%e f_0 = f_1 = 1, f_2 = 1+q^2, f_3 = 1+q^2+2q^3+q^4+q^6, so sequence begins 1; 1; 1,0,1; 1,0,1,2,1,0,1; ...

%p b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,

%p add(b(u-j, o+j-1)*x^(-u), j=1..u)+

%p add(b(u+j-1, o-j)*x^( o), j=1..o)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):

%p seq(T(n), n=0..10); # _Alois P. Heinz_, Apr 28 2018

%t qpoch[x_, n_] := Product[1-x*q^i, {i, 0, n-1}]; f[0]=1; f[n_] := f[n]=Together[Sum[ -(-1)^r q^Binomial[r, 2] qpoch[q^(n-r+1), r-1]*qpoch[q^(r+1), n-r]/qpoch[q, n-r] f[n-r], {r, 1, n}]]; Join@@Table[CoefficientList[f[n], q], {n, 0, 7}]

%Y Row sums give A000142.

%Y Cf. A052847.

%K nonn,tabf,easy

%O 0,9

%A _Dean Hickerson_, using information supplied by _Moshe Shmuel Newman_ and _Richard Stanley_, Mar 15 2003