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A122704
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a(n) = sum(k=0..n, 3^(n-k)*A123125(n, k) ).
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4
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1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216, 880405504, 17630351872, 385148108800, 9114999832576, 232311251144704, 6343764407375872, 184778982658539520
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OFFSET
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0,3
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COMMENTS
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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,...].
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REFERENCES
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T. J. Stieltjes, Sur quelques integrales definies et leur developpement en fractions continues. (Q. J. Math., London, 24, 1890, 370-382pp.); too
LXXVII, p.382, Stieltjes T.J. Oeuvres completes, tome 2, Noordhoff, 1918, 617p.
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LINKS
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Table of n, a(n) for n=0..16.
Eulerian polynomials.
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FORMULA
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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 DELEHAM, 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.
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PROG
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(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 */
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CROSSREFS
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Sequence in context: A112697 A113717 A124563 * A087547 A218678 A184942
Adjacent sequences: A122701 A122702 A122703 * A122705 A122706 A122707
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KEYWORD
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nonn
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AUTHOR
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Philippe DELEHAM, Oct 22 2006
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EXTENSIONS
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a(7) corrected (was 206672), a(n) extended, formula added Peter Luschny, Aug 03 2010
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STATUS
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approved
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