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A152550 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2), as a triangle read by rows.

%I A152550 #14 Feb 16 2025 08:33:09
%S A152550 1,1,3,2,12,16,16,5,55,110,170,180,130,70,14,273,728,1443,2145,2640,
%T A152550 2614,2200,1485,783,288,42,1428,4760,11312,20657,32032,42833,50477,
%U A152550 52934,49441,41069,29876,19019,10010,4158,1155,132,7752,31008,85272
%N A152550 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2), as a triangle read by rows.
%C A152550 LambertW satisfies: [LambertW(-2x)/(-2x)]^(1/2) = exp(x*LambertW(-2x)/(-2x)).
%H A152550 Paul D. Hanna, <a href="/A152550/b152550.txt">Rows 0 to 30 of the triangle, flattened.</a>
%H A152550 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a>.
%H A152550 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A152550 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 A152550 G.f.: A(x,q) = [(1/x)*Series_Reversion( x/e_q(x,q)^2 )]^(1/2) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
%F A152550 G.f. satisfies: A(x,q) = e_q( x*A(x,q)^2, q) and A( x/e_q(x,q)^2, q) = e_q(x,q).
%F A152550 G.f. at q=1: A(x,1) = (LambertW(-2*x)/(-2*x))^(1/2).
%F A152550 Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = (2*n+1)^(n-1).
%F A152550 Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = (2*n+1)^[(n-1)/2].
%F A152550 Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 1 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 1 for n>=1. - Vladeta Jovovic
%F A152550 Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} (2*n)!/(2*n-k+1)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs through all nonnegative integer solutions of e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - _Vladeta Jovovic_, Dec 04 2008
%e A152550 Triangle begins:
%e A152550   1;
%e A152550   1;
%e A152550   3,2;
%e A152550   12,16,16,5;
%e A152550   55,110,170,180,130,70,14;
%e A152550   273,728,1443,2145,2640,2614,2200,1485,783,288,42;
%e A152550   1428,4760,11312,20657,32032,42833,50477,52934,49441,41069,29876,19019,10010,4158,1155,132;
%e A152550   7752,31008,85272,181356,328440,521152,745416,969000,1159060,1278996,1307556,1238368,1085488,877240,650052,437164,262964,138320,60424,20592,4576,429;...
%e A152550 where row sums = (2*n+1)^(n-1) (A052750).
%e A152550 Row sums at q=-1 = (2*n+1)^[(n-1)/2] (A152551).
%e A152550 The generating function starts:
%e A152550 A(x,q) = 1 + x + (3 + 2*q)*x^2/faq(2,q) + (12 + 16*q + 16*q^2 + 5*q^3)*x^3/faq(3,q) + (55 + 110*q + 170*q^2 + 180*q^3 + 130*q^4 + 70*q^5 + 14*q^6)*x^4/faq(4,q) + ...
%e A152550 G.f. satisfies: A(x,q) = e_q( x*A(x,q)^2, q) where q-exponential series: e_q(x,q) = 1 + x + x^2/faq(2,q) + x^3/faq(3,q) +...+ x^n/faq(n,q) +...
%e A152550 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 A152550 Special cases.
%e A152550 q=0: A(x,0) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 +... (A001764)
%e A152550 q=1: A(x,1) = 1 + x + 5/2*x^2 + 49/6*x^3 + 729/24*x^4 + 14641/120*x^5 +...
%e A152550 q=2: A(x,2) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
%e A152550 q=3: A(x,3) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*x^5 +...
%o A152550 (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=sqrt(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 A152550 Cf. A052750 (row sums), A001764 (column 0), A000108 (right border), A152554.
%Y A152550 Cf. A152551 (q=-1), A152552 (q=2), A152553 (q=3).
%Y A152550 Cf. variants: A152290, A152555.
%K A152550 eigen,nonn,tabf
%O A152550 0,3
%A A152550 _Paul D. Hanna_, Dec 07 2008