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

1, 1, 3, 2, 12, 16, 16, 5, 55, 110, 170, 180, 130, 70, 14, 273, 728, 1443, 2145, 2640, 2614, 2200, 1485, 783, 288, 42, 1428, 4760, 11312, 20657, 32032, 42833, 50477, 52934, 49441, 41069, 29876, 19019, 10010, 4158, 1155, 132, 7752, 31008, 85272

Offset

0,3

Comments

LambertW satisfies: [LambertW(-2x)/(-2x)]^(1/2) = exp(x*LambertW(-2x)/(-2x)).

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 )]^(1/2) 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)^2, q) and A( x/e_q(x,q)^2, q) = e_q(x,q).

G.f. at q=1: A(x,1) = [LambertW(-2x)/(-2x)]^(1/2).

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

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

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

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). [From Vladeta Jovovic, Dec 04 2008]

Example

Triangle begins:
1;
1;
3,2;
12,16,16,5;
55,110,170,180,130,70,14;
273,728,1443,2145,2640,2614,2200,1485,783,288,42;
1428,4760,11312,20657,32032,42833,50477,52934,49441,41069,29876,19019,10010,4158,1155,132;
7752,31008,85272,181356,328440,521152,745416,969000,1159060,1278996,1307556,1238368,1085488,877240,650052,437164,262964,138320,60424,20592,4576,429;...
where row sums = (2*n+1)^(n-1) (A052750).
Row sums at q=-1 = (2*n+1)^[(n-1)/2] (A152551).
The generating function starts:
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) + ...
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) +...
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 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 +... (A001764)
q=1: A(x,1) = 1 + x + 5/2*x^2 + 49/6*x^3 + 729/24*x^4 + 14641/120*x^5 +...
q=2: A(x,2) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
q=3: A(x,3) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*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=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)}

Crossrefs

Cf. A052750 (row sums), A001764 (column 0), A000108 (right border), A152554.

Cf. A152551 (q=-1), A152552 (q=2), A152553 (q=3).

Cf. variants: A152290, A152555.

Sequence in context: A243660 A220883 A253246 * A114798 A167639 A113205

Adjacent sequences: A152547 A152548 A152549 * A152551 A152552 A152553

Keyword

eigen,nonn,tabf

Author

Paul D. Hanna, Dec 07 2008

Status

approved