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
KEYWORD
eigen,nonn,tabf
AUTHOR
Paul D. Hanna, Dec 02 2008
STATUS
approved