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A152555 Coefficients in a q-analog of the function LambertW(-2*x)/(-2*x), as a triangle read by rows.

%I A152555 #13 Feb 16 2025 08:33:09
%S A152555 1,2,7,5,30,42,42,14,143,297,462,495,363,198,42,728,2002,4004,6006,
%T A152555 7436,7436,6292,4290,2288,858,132,3876,13260,31824,58604,91364,122876,
%U A152555 145535,153361,143936,120185,87971,56329,29939,12584,3575,429,21318,87210
%N A152555 Coefficients in a q-analog of the function LambertW(-2*x)/(-2*x), as a triangle read by rows.
%H A152555 Paul D. Hanna, <a href="/A152555/b152555.txt">Rows 0 to 30 of the triangle, flattened.</a>
%H A152555 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a>.
%H A152555 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A152555 G.f.: A(x,q) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k)*q^k*x^n/faq(n,q), where faq(n,q) is the q-factorial of n.
%F A152555 G.f.: A(x,q) = (1/x)*Series_Reversion( x/e_q(x,q)^2 ) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
%F A152555 G.f. satisfies: A(x,q) = e_q( x*A(x,q), q)^2 and A( x/e_q(x,q)^2, q) = e_q(x,q)^2.
%F A152555 G.f. at q=1: A(x,1) = LambertW(-2*x)/(-2*x).
%F A152555 Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = 2*(2*n+2)^(n-1).
%F A152555 Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = 2*(2*n+2)^[(n-1)/2].
%F A152555 Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 2 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 2 for n>=1. - Vladeta Jovovic
%F A152555 Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} 2*(2*n+1)!/(2*n-k+2)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - _Vladeta Jovovic_, Dec 07 2008
%e A152555 Triangle begins:
%e A152555   1;
%e A152555   2;
%e A152555   7,5;
%e A152555   30,42,42,14;
%e A152555   143,297,462,495,363,198,42;
%e A152555   728,2002,4004,6006,7436,7436,6292,4290,2288,858,132;
%e A152555   3876,13260,31824,58604,91364,122876,145535,153361,143936,120185,87971,56329,29939,12584,3575,429;
%e A152555   21318,87210,242250,519384,945744,1508070,2165664,2826420,3392520,3756626,3853322,3662106,3221330,2613240,1944324,1313760,794614,420784,185640,64090,14586,1430;...
%e A152555 where row sums = 2*(2*n+2)^(n-1) (A097629).
%e A152555 Row sums at q=-1 = 2*(2*n+2)^[(n-1)/2] (A152556).
%e A152555 The generating function starts:
%e A152555 A(x,q) = 1 + 2*x + (7 + 5*q)*x^2/faq(2,q) + (30 + 42*q + 42*q^2 + 14*q^3)*x^3/faq(3,q) + (143 + 297*q + 462*q^2 + 495*q^3 + 363*q^4 + 198*q^5 + 42*q^6)*x^4/faq(4,q) + ...
%e A152555 The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1): faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2), faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...
%e A152555 Special cases.
%e A152555 q=0: A(x,0) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... (A006013)
%e A152555 q=1: A(x,1) = 1 + 2*x + 12/2*x^2 + 128/6*x^3 + 2000/24*x^4 + 41472/120*x^5 +...
%e A152555 q=2: A(x,2) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
%e A152555 q=3: A(x,3) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*x^5 +...
%o A152555 (PARI) {T(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); polcoeff(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}
%Y A152555 Cf. A097629 (row sums), A006013 (column 0), A000108 (right border), A152559.
%Y A152555 Cf. A152556 (q=-1), A152557 (q=2), A152558 (q=3).
%Y A152555 Cf. variants: A152290, A152550.
%K A152555 eigen,nonn,tabf
%O A152555 0,2
%A A152555 _Paul D. Hanna_, Dec 07 2008