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 A210246 Polylogarithm li(-n,-1/3) multiplied by (4^(n+1))/3. 35
 1, -1, -2, 2, 40, 104, -1232, -13168, 16000, 1483904, 9695488, -151161088, -3287997440, 146760704, 866038110208, 10263094740992, -169941494497280, -6324725967978496, -15215847186563072, 2895126258819203072, 54295929047166484480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Given n, consider the series s(n) = li(-n,-1/3) = SUM((-1)^k (k^n)/3^k) for k=0,1,2,... . Then a(n)=s(n)*(4^(n+1))/3. For more details, see A212846. LINKS Stanislav Sykora, Table of n, a(n) for n = 0..100 OEIS-Wiki, Eulerian polynomials S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), Stan's Library Vol. I, April 2006, updated March 2012. See Eq.(29). Eric W. Weisstein, MathWorld: Polylogarithm FORMULA Recurrence: s(n+1)=(-1/4)*SUM(C(n+1,i)*s(i)), where i=0,1,2,...,n, C(n,m) are binomial coefficients, and the starting value is s(0)=SUM((-1/3)^k)=3/4. From Peter Bala, Mar 12 2013: (Start) E.g.f.: A(x) = 4/(3 + exp(4*x)) = 1 - x - 2*x^2/2! + 2*x^3/3! + 40*x^4/4! + .... The compositional inverse (A(-x) - 1)^(-1) = x + 2*x^2/2 + 7*x^3/3 + 20*x^4/4 + 61*x^5/5 + ... is the logarithmic generating function for A015518. Recurrence equation: a(n+1) = -4*a(n) + 3*sum {k = 0..n} binomial(n,k)*a(k)*a(n-k), with a(0) = 1. (End) G.f.: 1 + x/Q(0), where Q(k) = 2*x*(k+1) - 1 + 3*x^2*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 22 2013 G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 3*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013 E.g.f.: 2 - W(0), where W(k) = 1 + x/( 4*k+1 - x/( 1 + 4*x/( 4*k+3 - 4*x/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2014 EXAMPLE a(5) = polylog(-5,-1/3)*4^6/3 = 104. MAPLE seq(add((-1)^(n-k)*combinat[eulerian1](n, k)*3^k, k=0..n), n=0..20); # Peter Luschny, Apr 21 2013 MATHEMATICA Table[PolyLog[-n, -1/3] (4^(n+1))/3, {n, 30}] (* T. D. Noe, Mar 23 2012 *) a[ n_] := If[ n < 1, Boole[n == 0], PolyLog[ -n, -1/3] 4^(n + 1) / 3]; (* Michael Somos, Nov 01 2014 *) PROG (PARI) /* See in A212846, run limnpq(nmax, 1, 3) */ (PARI) x='x+O('x^66); Vec(serlaplace( 4/(3+exp(4*x)) )) \\ Joerg Arndt, Apr 21 2013 CROSSREFS Similar to A210244. Cf. A210247 (sign changes). Cf. A212846 (li(-n,-1/2)), A212847 (li(-n,-2/3)). CF. A213127 through A213157. Sequence in context: A001121 A202711 A222668 * A116567 A215049 A087541 Adjacent sequences:  A210243 A210244 A210245 * A210247 A210248 A210249 KEYWORD sign AUTHOR Stanislav Sykora, Mar 19 2012 STATUS approved

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