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 A003727 Expansion of e.g.f. exp(x * cosh(x)). (Formerly M3462) 4
 1, 1, 1, 4, 13, 36, 181, 848, 3865, 23824, 140521, 871872, 6324517, 44942912, 344747677, 2860930816, 23853473329, 213856723200, 1996865965009, 19099352929280, 193406280000061, 2010469524579328, 21615227339380357, 242177953175506944 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..500 Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011-2013. FORMULA a(n) = Sum_{k=1..n} (if n=k then n! otherwise (1/2)^k*Sum_{i=0..k} binomial(n,k)* binomial(k,i)*(k-2*i)^(n-k)), n>0. - Vladimir Kruchinin, Aug 22 2010 a(n) ~ exp(r*cosh(r)-n) * n^n / (r^n * sqrt(3+(r*(r^2-2)*cosh(r))/n)), where r is the root of the equation r*(cosh(r)+r*sinh(r)) = n. - Vaclav Kotesovec, Aug 05 2014 a(n)^(1/n) ~ n*exp(1/(2*LambertW(sqrt(n/2)))-1) / (2*LambertW(sqrt(n/2))). - Vaclav Kotesovec, Aug 05 2014 MATHEMATICA CoefficientList[Series[E^(x*Cosh[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 05 2014 *) Table[Sum[BellY[n, k, Mod[Range[n], 2] Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *) PROG (Maxima) a(n):=sum(if n=k then n! else 1/2^k*sum(binomial(n, k)*binomial(k, i)*(k-2*i)^(n-k), i, 0, k), k, 1, n); /* Vladimir Kruchinin, Aug 22 2010 */ (PARI) x='x+O('x^66); Vec(serlaplace(exp( x * cosh(x) ))) /* Joerg Arndt, Sep 14 2012 */ (MAGMA) m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Cosh(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; \\ G. C. Greubel, Sep 09 2018 CROSSREFS Cf. A009233, A191509. Sequence in context: A067635 A222425 A222189 * A103082 A279111 A299111 Adjacent sequences:  A003724 A003725 A003726 * A003728 A003729 A003730 KEYWORD nonn AUTHOR EXTENSIONS Extended and formatted by Olivier Gérard, Mar 15 1997 STATUS approved

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Last modified October 16 13:08 EDT 2019. Contains 328063 sequences. (Running on oeis4.)