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%I #7 Feb 21 2021 10:55:23
%S 1,1,3,1,1,8,24,30,20,7,1,1,15,99,368,841,1302,1435,1149,668,276,77,
%T 13,1,1,24,268,1828,8446,28050,70378,138204,217525,278663,293217,
%U 254462,182077,106926,51053,19515,5831,1312,209,21,1,1,35,585,6180,46080,257302,1119568,3916826,11290146
%N Irregular table: the n-th row polynomial is given by the formal power series expansion of Sum_{k >= 0} (1 + q)^(n*k + n^2)*Product_{j = 1..k} (1 - (1 + q)^(2*j-1)), n >= 1.
%C For n = 1,2,3,..., it can be shown that the formal power series Sum_{k >= 0} (1 + q)^(n*k)*Product_{i = 1..k} (1 - (1 + q)^(2*i-1)) is a rational function of q of the form R(n,q)/(1 + q)^(n^2), where R(n,q) is a polynomial of degree n*(n-1) in q. The first few values are R(1,q) = 1, R(2,q) = 1 + 3*q + q^2 and R(3,q) = 1 + 8*q + 24*q^2 + 30*q^3 + 20*q^4 + 7*q^5 + q^6. The present table lists the coefficients of the polynomials R(n,q) for n >= 1.
%C Conjecture: For fixed integers q and N, the sequence mod(R(n,q),N)n>=1 is either purely periodic or periodic with a pre-period 1.
%F n-th row polynomial: R(n,q) = Sum_{k = 0..n-1} x^(k^2) * ( Product_{j = k+1..n-1} x^(2*j) - 1 ), where x = 1 + q.
%F In the q-adic topology, Limit_{n -> oo} R(2*n+1, q - 1) = - Limit_{n -> oo} R(2*n, q - 1) = Product_{n >= 1} 1 - q^n.
%e Table begins
%e n\k 0 1 2 3 4 5 6 7 8
%e ------------------------------------------------------------------------
%e 1 1,
%e 2 1, 3, 1,
%e 3 1, 8, 24, 30, 20, 7, 1,
%e 4 1, 15, 99, 368, 841, 1302, 1435, 1149, 668, ...
%e 5 1, 24, 268, 1828, 8446, 28050, 70378, 138204, 217525, ...
%e 6 1, 35, 585, 6180, 46080, 257302, 1119568, 3916826, 11290146, ...
%e ...
%p row := n -> add( (1+q)^(k^2)*mul((1+q)^(2*j)-1, j = k+1..n-1), k = 0..n-1 ):
%p seq(seq(coeff(row(n), q, k), k = 0..n*(n-1)), n = 1..10);
%Y Row sums A340883. Cf. A340880.
%K nonn,tabf,easy
%O 1,3
%A _Peter Bala_, Feb 16 2021