A152290 Coefficients in a q-analog of the LambertW function, as a triangle read by rows.

1, 1, 2, 1, 5, 5, 5, 1, 14, 21, 31, 30, 19, 9, 1, 42, 84, 154, 210, 245, 217, 175, 105, 49, 14, 1, 132, 330, 708, 1176, 1722, 2148, 2386, 2358, 2080, 1618, 1086, 644, 294, 104, 20, 1, 429, 1287, 3135, 6006, 10164, 15093, 20496, 25188, 28770, 30225, 29511, 26571

Offset

0,3

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) ) 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) and A( x/e_q(x,q), q) = e_q(x,q).

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

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

Row sums at q=-1: Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = (n+1)^[(n-1)/2] (A152291).

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 Paul D. Hanna, Dec 04 2008]

Sum_{k=0..n*(n-1)/2} T(n,k)*q^k = Sum_{pi} n!/(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 05 2008]

Sum_{k=0..[n/2]} T(n, n*k) = (1/n)*Sum_{d|n} phi(n/d)*(n+1)^(d-1), for n>0, with a(0)=1. [From Paul D. Hanna, Jul 18 2013]

Sum_{k=0..[n/2]} T(n, n*k) = A121774(n), the number of n-bead necklaces with n+1 colors, divided by (n+1). [From Paul D. Hanna, Jul 18 2013]

Example

Triangle, with columns k=0..n(n-1)/2 for row n>=0, begins:
1;
1;
2, 1;
5, 5, 5, 1;
14, 21, 31, 30, 19, 9, 1;
42, 84, 154, 210, 245, 217, 175, 105, 49, 14, 1;
132, 330, 708, 1176, 1722, 2148, 2386, 2358, 2080, 1618, 1086, 644, 294, 104, 20, 1;
429, 1287, 3135, 6006, 10164, 15093, 20496, 25188, 28770, 30225, 29511, 26571, 22161, 16926, 11832, 7392, 4089, 1932, 714, 195, 27, 1;...
where row sums = (n+1)^(n-1) and column 0 is A000108 (Catalan numbers).
Row sums at q=-1 = (n+1)^[(n-1)/2] (A152291): [1,1,1,4,5,36,49,512,729,...].
The generating function starts:
A(x,q) = 1 + x + (2 + q)*x^2/faq(2,q) + (5 + 5*q + 5*q^2 + q^3)*x^3/faq(3,q) + (14 + 21*q + 31*q^2 + 30*q^3 + 19*q^4 + 9*q^5 + q^6)*x^4/faq(4,q) + ...
G.f. satisfies: A(x,q) = e_q( x*A(x,q), q), where the q-exponential series e_q(x,q) begins:
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 of g.f.:
q=0: A(x,0) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (Catalan)
q=1: A(x,1) = 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 +...= LambertW(-x)/(-x)
q=2: A(x,2) = 1 + x + 4/3*x^2 + 43/21*x^3 + 1076/315*x^4 + 58746/9765*x^5 +...
q=-1: Can A(x,-1) be defined? See A152291.

Prog

(PARI) /* G.f.: LambertW_q(x, q) = (1/x)*Series_Reversion( x/e_q(x, q) ): */
{T(n, k)=local(e_q=1+sum(j=1, n, x^j/prod(i=1, j, (q^i-1)/(q-1))), LW_q=serreverse(x/e_q+x^2*O(x^n))/x); polcoeff(polcoeff(LW_q+x*O(x^n), n, x)*prod(i=1, n, (q^i-1)/(q-1))+q*O(q^k), k, q)}
for(n=0, 8, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A152291 (q=-1), A000272 (row sums), A000108 (column 0), A002054 (column 1).

Cf. A152282 (q=2), A152283 (q=3).

Cf. A121774.

Sequence in context: A124226 A248797 A193536 * A248699 A032006 A167158

Adjacent sequences: A152287 A152288 A152289 * A152291 A152292 A152293

Keyword

eigen,nonn,tabf

Author

Paul D. Hanna, Dec 02 2008

Status

approved