login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A152283 Coefficients in a q-analog of the LambertW function at q=3: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,3) where faq(n,q) = q-factorial of n. 2

%I #5 Jul 10 2015 19:37:43

%S 1,1,5,92,5621,1093236,663362421,1242109529088,7129029760138649,

%T 124860091946887218320,6652206059042029394600021,

%U 1075572123264132205051327968256,526826946994724781414669857330392909

%N Coefficients in a q-analog of the LambertW function at q=3: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,3) where faq(n,q) = q-factorial of n.

%F G.f. satisfies: A(x) = e_q( x*A(x), 3) and A( x/e_q(x,3) ) = e_q(x,3) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.

%F G.f.: A(x) = (1/x)*Series_Reversion( x/e_q(x,3) ).

%F a(n) = Sum_{k=0..n(n-1)/2} A152290(n,k)*3^k.

%F a(n) = faq(n,3)*Sum_{pi} n!/((n-k+1)!*Product_{i=1..n} (e(i)!*faq(i,3)^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 03 2008]

%e G.f.: A(x) = 1 + x + 5/4*x^2 + 92/52*x^3 + 5621/2080*x^4 + 1093236/251680*x^5 +...

%e G.f. satisfies: A(x) = e_q( x*A(x), 3) where the q-exponential series is:

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

%e e_q(x,3) = 1 + x + x^2/4 + x^3/52 + x^4/2080 + x^5/251680 +...

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

%o (PARI) {a(n,q=3)=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(LW_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))}

%Y Cf. A152290, A152282 (q=2).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 02 2008

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)