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A081285 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,</span> where (x)<span class="maths">_n</span> is the q-Pochhammer symbol, defined to be <span class="maths">Product{i=0..n-1} (1-x q^i).

%I A081285 #34 Feb 16 2025 08:32:48
%S A081285 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 A081285 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 A081285 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 A081285 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 A081285 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 A081285 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 A081285 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.
%H A081285 Alois P. Heinz, <a href="/A081285/b081285.txt">Rows n = 0..40, flattened</a>
%H A081285 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 A081285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol.</a>
%F A081285 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 A081285 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 A081285 b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
%p A081285       add(b(u-j, o+j-1)*x^(-u), j=1..u)+
%p A081285       add(b(u+j-1, o-j)*x^( o), j=1..o)))
%p A081285     end:
%p A081285 T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):
%p A081285 seq(T(n), n=0..10);  # _Alois P. Heinz_, Apr 28 2018
%t A081285 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 A081285 Row sums give A000142.
%Y A081285 Cf. A052847.
%K A081285 nonn,tabf,easy
%O A081285 0,9
%A A081285 _Dean Hickerson_, using information supplied by _Moshe Shmuel Newman_ and _Richard Stanley_, Mar 15 2003