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A122704 a(n) = sum( k=0..n, 3^(n-k)*A123125(n, k) ). 5
1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216, 880405504, 17630351872, 385148108800, 9114999832576, 232311251144704, 6343764407375872, 184778982658539520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n+1) = [1,4,22,160,1456,...] is the first Eulerian transform of A000244 (powers of 3), it is also the Stirling transform of A080599(n+1) = [1,3,12,66,450,...].

REFERENCES

T. J. Stieltjes, Sur quelques integrales definies et leur developpement en fractions continues, LXXVII, p.382, Stieltjes T.J. Oeuvres completes, tome 2, Noordhoff, 1918, 617p.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..100

T. J. Stieltjes, Sur quelques intégrales definies et leur développement en fractions continues, Q. J. Math., London, 24, 1890, pp. 370-382.

OEIS Wiki, Eulerian polynomials.

Eric Weisstein's MathWorld, Polylogarithm.

FORMULA

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1-2*k*x). - Paul D. Hanna, Jul 20 2011

a(n) = sum( k=0..n, A131689(n,k)*2^(n-k) ). - Philippe Deléham, Oct 09 2007

a(n) = A_{n}(3) where A_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010

E.g.f.: (exp(x) - 2*cosh(x))/(2*exp(x) - 3*cosh(x)) =1 + x/(U(0)-x) where U(k)=  4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012

G.f.: 1 + x/G(0) where G(k) = 1 - x*2*(2*k+2) + x^2*(k+1)*(k+2)*(1-2^2)/G(k+1); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013

a(n) ~ n!/3 * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013

G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 3*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013

a(n) = sum( k>=0, 2^(n+1)*k^n/3^(k+1) ). - Vaclav Kotesovec, Nov 28 2013

a(n) = 2^n*log(3)* integral {x >= 0} (floor(x))^n * 3^(-x) dx. - Peter Bala, Feb 14 2015

From Karol A. Penson, Sep 04 2015: (Start)

E.g.f.: 2/(3-exp(2*x)).

Special values of the generalized hypergeometric function n_F_(n-1):

a(n) = (2^(n+1)/9) * hypergeom([2,2,..2],[1,1,..1],1/3), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.

Example: a(4) = (2^5/9) * hypergeom([2,2,2,2],[1,1,1],1/3) = 16. (End)

a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3)))/3, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015

MATHEMATICA

CoefficientList[Series[(Exp[x]-2*Cosh[x])/(2*Exp[x]-3*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)

Table[Sum[2^(n+1)*k^n/3^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2013 *)

Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[3]] + PolyLog[-n, -Sqrt[3]])/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1-2*k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */

CROSSREFS

Cf. A076726.

Sequence in context: A112697 A113717 A124563 * A087547 A218678 A184942

Adjacent sequences:  A122701 A122702 A122703 * A122705 A122706 A122707

KEYWORD

nonn,easy

AUTHOR

Philippe Deléham, Oct 22 2006

EXTENSIONS

a(7) corrected (was 206672), a(n) extended, formula added Peter Luschny, Aug 03 2010

STATUS

approved

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Last modified August 21 02:25 EDT 2017. Contains 290855 sequences.