A122704 a(n) = Sum_{k=0..n} 3^(n-k)*A123125(n, k).

1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216, 880405504, 17630351872, 385148108800, 9114999832576, 232311251144704, 6343764407375872, 184778982658539520, 5718564661248065536, 187389427488113557504, 6481629887083387420672, 235993351028007334051840

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,...].

Links

Alois P. Heinz, Table of n, a(n) for n = 0..409 (first 101 terms from G. C. Greubel)

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

T. J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, LXXVII, p.382, Stieltjes T.J. Oeuvres complètes, tome 2, Noordhoff, 1918, 617p.

OEIS Wiki, Eulerian polynomials.

Eric Weisstein's World of Mathematics, 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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 2^(k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020

a(n) = 2^n*F_{n}(1/2), where F_{n}(x) is the Fubini polynomial. This is another way to state the above formula from Ilya Gutkovskiy. - Peter Luschny, May 21 2021

a(n+1) = -2*a(n) + 3*Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). - Michael Somos, Jun 05 2021

a(n) = (-2)^(n + 1)*PolyLog(-n, 3)/3. - Peter Luschny, Aug 20 2021

Example

G.f. = 1 + x + 4*x^2 + 22*x^3 + 160*x^4 + 1456*x^5 + 15904*x^6 + ... - Michael Somos, Jun 05 2021

Maple

# From Peter Luschny, Jun 27 2019: (Start)
seq(subs(x=3, add(combinat:-eulerian1(n, k)*x^k, k=0..n)), n=0..20);
# Alternative:
gf := -2/(exp(2*x) - 3): series(gf, x, 21): seq(n!*coeff(%, x, n), n=0..20);
# (End)
# Or via the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
a := n -> 2^n*subs(x = 1/2, F(n)):
seq(a(n), n = 0..20); # Peter Luschny, May 21 2021
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021

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 *)
Table[Sum[StirlingS2[n, k]*2^(n-k)*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 13 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[ n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k]*3^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)
a[n_] := (-2)^(n + 1) PolyLog[-n, 3] / 3;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Aug 20 2021 *)

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 */
(PARI) {a(n) = if(n<0, 0, n!*polcoeff( 2/(3 - exp(2*x + x*O(x^n))), n))}; /* Michael Somos, Jun 05 2021 */

Crossrefs

Cf. A076726.

Sequence in context: A327994 A113717 A124563 * A087547 A218678 A346968

Adjacent sequences: A122701 A122702 A122703 * A122705 A122706 A122707

Keyword

nonn,easy

Author

Philippe Deléham, Oct 22 2006

Extensions

a(7) corrected (was 206672) and more terms from Peter Luschny, Aug 03 2010

More terms from Vaclav Kotesovec, Jul 13 2018

Status

approved