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%I #93 Aug 20 2021 12:51:50
%S 1,1,4,22,160,1456,15904,202672,2951680,48361216,880405504,
%T 17630351872,385148108800,9114999832576,232311251144704,
%U 6343764407375872,184778982658539520,5718564661248065536,187389427488113557504,6481629887083387420672,235993351028007334051840
%N a(n) = Sum_{k=0..n} 3^(n-k)*A123125(n, k).
%C 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,...].
%H Alois P. Heinz, <a href="/A122704/b122704.txt">Table of n, a(n) for n = 0..409</a> (first 101 terms from G. C. Greubel)
%H T. J. Stieltjes, <a href="https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:[388],%22view%22:%22info%22}">Sur quelques intégrales définies et leur développement en fractions continues</a>, Q. J. Math., London, 24, 1890, pp. 370-382.
%H T. J. Stieltjes, <a href="https://archive.org/details/oeuvresthomasja02stierich/page/n387/mode/2up">Sur quelques intégrales définies et leur développement en fractions continues</a>, LXXVII, p.382, Stieltjes T.J. Oeuvres complètes, tome 2, Noordhoff, 1918, 617p.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1-2*k*x). - _Paul D. Hanna_, Jul 20 2011
%F a(n) = Sum_{k=0..n} A131689(n,k)*2^(n-k). - _Philippe Deléham_, Oct 09 2007
%F a(n) = A_{n}(3) where A_{n}(x) are the Eulerian polynomials. - _Peter Luschny_, Aug 03 2010
%F 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
%F 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
%F a(n) ~ n!/3 * (2/log(3))^(n+1). - _Vaclav Kotesovec_, Jun 24 2013
%F 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
%F a(n) = Sum_{k>=0} 2^(n+1)*k^n/3^(k+1). - _Vaclav Kotesovec_, Nov 28 2013
%F a(n) = 2^n*log(3)* Integral_{x >= 0} (floor(x))^n * 3^(-x) dx. - _Peter Bala_, Feb 14 2015
%F From _Karol A. Penson_, Sep 04 2015: (Start)
%F E.g.f.: 2/(3-exp(2*x)).
%F Special values of the generalized hypergeometric function n_F_(n-1):
%F 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.
%F Example: a(4) = (2^5/9) * hypergeom([2,2,2,2],[1,1,1],1/3) = 16. (End)
%F 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
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 2^(k - 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 16 2020
%F 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
%F a(n+1) = -2*a(n) + 3*Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). - _Michael Somos_, Jun 05 2021
%F a(n) = (-2)^(n + 1)*PolyLog(-n, 3)/3. - _Peter Luschny_, Aug 20 2021
%e G.f. = 1 + x + 4*x^2 + 22*x^3 + 160*x^4 + 1456*x^5 + 15904*x^6 + ... - _Michael Somos_, Jun 05 2021
%p # From _Peter Luschny_, Jun 27 2019: (Start)
%p seq(subs(x=3, add(combinat:-eulerian1(n,k)*x^k, k=0..n)), n=0..20);
%p # Alternative:
%p gf := -2/(exp(2*x) - 3): series(gf, x, 21): seq(n!*coeff(%, x, n), n=0..20);
%p # (End)
%p # Or via the recurrence of the Fubini polynomials:
%p F := proc(n) option remember; if n = 0 then return 1 fi;
%p expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
%p a := n -> 2^n*subs(x = 1/2, F(n)):
%p seq(a(n), n = 0..20); # _Peter Luschny_, May 21 2021
%p # next Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p a(n-j)*binomial(n, j)*2^(j-1), j=1..n))
%p end:
%p seq(a(n), n=0..23); # _Alois P. Heinz_, May 30 2021
%t 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 *)
%t Table[Sum[2^(n+1)*k^n/3^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 28 2013 *)
%t Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[3]] + PolyLog[-n, -Sqrt[3]])/3, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 31 2015 *)
%t Table[Sum[StirlingS2[n, k]*2^(n-k)*k!, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 13 2018 *)
%t 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_ *)
%t a[n_] := (-2)^(n + 1) PolyLog[-n, 3] / 3;
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Aug 20 2021 *)
%o (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 */
%o (PARI) {a(n) = if(n<0, 0, n!*polcoeff( 2/(3 - exp(2*x + x*O(x^n))), n))}; /* _Michael Somos_, Jun 05 2021 */
%Y Cf. A076726.
%K nonn,easy
%O 0,3
%A _Philippe Deléham_, Oct 22 2006
%E a(7) corrected (was 206672) and more terms from _Peter Luschny_, Aug 03 2010
%E More terms from _Vaclav Kotesovec_, Jul 13 2018
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