0,3

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

Eric Weisstein, q-Exponential Function from MathWorld.

Eric Weisstein, q-Factorial from MathWorld.

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 (pauldhanna(AT)juno.com), 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 (vladeta(AT)eunet.yu), Dec 05 2008]

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) = 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.

(PARI) /* for(n=0, 7, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print("")) */ {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)}

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

Cf. A152282 (q=2), A152283 (q=3). [From Paul D. Hanna (pauldhanna(AT)juno.com), Dec 04 2008]

Sequence in context: A058116 A058118 A124226 this_sequence A032006 A167158 A074392

Adjacent sequences: A152287 A152288 A152289 this_sequence A152291 A152292 A152293

eigen,nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 02 2008