A152555 Coefficients in a q-analog of the function LambertW(-2*x)/(-2*x), as a triangle read by rows.
1, 2, 7, 5, 30, 42, 42, 14, 143, 297, 462, 495, 363, 198, 42, 728, 2002, 4004, 6006, 7436, 7436, 6292, 4290, 2288, 858, 132, 3876, 13260, 31824, 58604, 91364, 122876, 145535, 153361, 143936, 120185, 87971, 56329, 29939, 12584, 3575, 429, 21318, 87210
Offset: 0
Examples
Triangle begins:
1;
2;
7,5;
30,42,42,14;
143,297,462,495,363,198,42;
728,2002,4004,6006,7436,7436,6292,4290,2288,858,132;
3876,13260,31824,58604,91364,122876,145535,153361,143936,120185,87971,56329,29939,12584,3575,429;
21318,87210,242250,519384,945744,1508070,2165664,2826420,3392520,3756626,3853322,3662106,3221330,2613240,1944324,1313760,794614,420784,185640,64090,14586,1430;...
where row sums = 2*(2*n+2)^(n-1) (A097629).
Row sums at q=-1 = 2*(2*n+2)^[(n-1)/2] (A152556).
The generating function starts:
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) + ...
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), ...
Special cases.
q=0: A(x,0) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... (A006013)
q=1: A(x,1) = 1 + 2*x + 12/2*x^2 + 128/6*x^3 + 2000/24*x^4 + 41472/120*x^5 +...
q=2: A(x,2) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
q=3: A(x,3) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*x^5 +...
Links
- Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
- Eric Weisstein's World of Mathematics, q-Exponential Function.
- Eric Weisstein's World of Mathematics, q-Factorial.
Crossrefs
Programs
-
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)}
Formula
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.
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.
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.
G.f. at q=1: A(x,1) = LambertW(-2*x)/(-2*x).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = 2*(2*n+2)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = 2*(2*n+2)^[(n-1)/2].
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
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
Comments