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

Author

Paul D. Hanna, Dec 02 2008

Keywords

• eigen
• nonn
• tabf

Comments

T(n,k) is the number of parking functions of length n with k inversions. - Kyle Celano, Aug 18 2025

Examples

			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.
		

Links

• Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
• Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025.
• Eric Weisstein's World of Mathematics, q-Exponential Function.
• Eric Weisstein's World of Mathematics, q-Factorial.

Crossrefs

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

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

Cf. A121774.

Programs

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(""))

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(2*Pi*I*k/n) = 1 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 1 for n>=1. - 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). - 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. - 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). - Paul D. Hanna, Jul 18 2013