A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), 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

0,2

Links

Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.

Eric Weisstein, q-Exponential Function from MathWorld.

Eric Weisstein, q-Factorial from MathWorld.

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(-2x)/(-2x).

Row sums at q=+1: Sum_{k=0..n(n-1)/2} T(n,k) = 2*(2n+2)^(n-1).

Row sums at q=-1: Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = 2*(2n+2)^[(n-1)/2].

Sum_{k=0..n(n-1)/2} T(n,k)*exp(2Pi*I*k/n)) = 2 for n>=1; i.e., the n-th row sum at q = exp(2Pi*I/n), the n-th root of unity, equals 2 for n>=1. [From 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). [From Vladeta Jovovic, Dec 07 2008]

Example

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 +...

Prog

(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)}

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.

Sequence in context: A100759 A205448 A180510 * A343780 A094360 A335594

Adjacent sequences: A152552 A152553 A152554 * A152556 A152557 A152558

Keyword

eigen,nonn,tabf

Author

Paul D. Hanna, Dec 07 2008

Status

approved