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

Author

Paul D. Hanna, Dec 07 2008

Keywords

• eigen
• nonn
• tabf

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

Cf. A097629 (row sums), A006013 (column 0), A000108 (right border), A152559.

Cf. A152556 (q=-1), A152557 (q=2), A152558 (q=3).

Cf. variants: A152290, A152550.

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