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 A152399 Log of the q-exponential of x, e_q(x,q), evaluated at q=-x. 1
 1, 1, 4, 9, 16, 22, 29, 49, 94, 156, 221, 318, 521, 883, 1429, 2257, 3605, 5836, 9463, 15264, 24539, 39579, 64148, 103990, 168141, 271623, 439276, 711055, 1150750, 1861287, 3010318, 4870449, 7881944, 12754455, 20635589, 33385764, 54018447 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The g.f.s for this sequence illustrates the following formula: log(e_q(x,q)) = Sum_{n>=1} (1-q)^n/(1-q^n)*x^n/n, where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential of x and faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n. LINKS Table of n, a(n) for n=1..37. Eric Weisstein, q-Exponential Function from MathWorld. Eric Weisstein, q-Factorial from MathWorld. FORMULA L.g.f.: log(e_q(x,-x)) = log(Sum_{n>=0} x^n/[Product_{k=1..n} (1-(-x)^k)/(1+x)]). L.g.f.: log(e_q(x,-x)) = Sum_{n>=1} x^n*(1+x)^n/(1-(-x)^n)/n. EXAMPLE L.g.f.: log(e_q(x,-x)) = x + x^2/2 + 4*x^3/3 + 9*x^4/4 + 16*x^5/5 + 22*x^6/6 +... e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 17*x^7 +... (A152398). PROG (PARI) a(n)=n*polcoeff(log(sum(k=0, n, x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n)))), n) (PARI) a(n)=polcoeff(sum(k=1, n, x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n), n) CROSSREFS Cf. A152398 (e_q(x, -x)). Sequence in context: A313348 A313349 A313350 * A022822 A001639 A309138 Adjacent sequences: A152396 A152397 A152398 * A152400 A152401 A152402 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 16 2008 STATUS approved

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Last modified September 27 05:14 EDT 2023. Contains 365674 sequences. (Running on oeis4.)