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A152290 Coefficients in a q-analog of the LambertW function, as a triangle read by rows. 9

%I

%S 1,1,2,1,5,5,5,1,14,21,31,30,19,9,1,42,84,154,210,245,217,175,105,49,

%T 14,1,132,330,708,1176,1722,2148,2386,2358,2080,1618,1086,644,294,104,

%U 20,1,429,1287,3135,6006,10164,15093,20496,25188,28770,30225,29511,26571

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

%H Paul D. Hanna, <a href="/A152290/b152290.txt">Rows 0 to 30 of the triangle, flattened.</a>

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a> from MathWorld.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a> from MathWorld.

%F 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.

%F 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.

%F 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).

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

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

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

%F 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]

%F 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]

%F 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]

%F 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]

%e Triangle, with columns k=0..n(n-1)/2 for row n>=0, begins:

%e 1;

%e 1;

%e 2, 1;

%e 5, 5, 5, 1;

%e 14, 21, 31, 30, 19, 9, 1;

%e 42, 84, 154, 210, 245, 217, 175, 105, 49, 14, 1;

%e 132, 330, 708, 1176, 1722, 2148, 2386, 2358, 2080, 1618, 1086, 644, 294, 104, 20, 1;

%e 429, 1287, 3135, 6006, 10164, 15093, 20496, 25188, 28770, 30225, 29511, 26571, 22161, 16926, 11832, 7392, 4089, 1932, 714, 195, 27, 1;...

%e where row sums = (n+1)^(n-1) and column 0 is A000108 (Catalan numbers).

%e Row sums at q=-1 = (n+1)^[(n-1)/2] (A152291): [1,1,1,4,5,36,49,512,729,...].

%e The generating function starts:

%e 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) + ...

%e G.f. satisfies: A(x,q) = e_q( x*A(x,q), q), where the q-exponential series e_q(x,q) begins:

%e e_q(x,q) = 1 + x + x^2/faq(2,q) + x^3/faq(3,q) +...+ x^n/faq(n,q) +...

%e The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1):

%e faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2),

%e faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...

%e Special cases of g.f.:

%e q=0: A(x,0) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (Catalan)

%e q=1: A(x,1) = 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 +...= LambertW(-x)/(-x)

%e q=2: A(x,2) = 1 + x + 4/3*x^2 + 43/21*x^3 + 1076/315*x^4 + 58746/9765*x^5 +...

%e q=-1: Can A(x,-1) be defined? See A152291.

%o (PARI) /* G.f.: LambertW_q(x,q) = (1/x)*Series_Reversion( x/e_q(x,q) ): */

%o {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)}

%o for(n=0,8,for(k=0,n*(n-1)/2,print1(T(n,k),","));print(""))

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

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

%Y Cf. A121774.

%K eigen,nonn,tabf

%O 0,3

%A _Paul D. Hanna_, Dec 02 2008

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Last modified April 17 21:53 EDT 2021. Contains 343071 sequences. (Running on oeis4.)