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 A111884 E.g.f.: exp(x/(1+x)). 23
 1, 1, -1, 1, 1, -19, 151, -1091, 7841, -56519, 396271, -2442439, 7701409, 145269541, -4833158329, 104056218421, -2002667085119, 37109187217649, -679877731030049, 12440309297451121, -227773259993414719, 4155839606711748061, -74724654677947488521, 1293162252850914402221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row sums of triangle A111596. With different signs see A066668. From Peter Bala, Aug 15 2022: (Start) The congruence a(n+k) == a(n) (mod k) holds for all n and k. It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with period dividing k. For example, taken modulo 10 the sequence becomes [1, 1, 9, 1, 1, 1, 1, 9, 1, 1, ...], a purely periodic sequence with period 5. More generally, the same property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1. (End) LINKS Seiichi Manyama, Table of n, a(n) for n = 0..450 Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4. Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6. Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20. Paul Barry, Combinatorial Polynomials as Moments, Hankel Transforms, and Exponential Riordan Arrays, J. Int. Seq. 14 (2011) 11.6.7, example 10. A. Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2 FORMULA E.g.f.: exp(x/(1+x)). From Sergei N. Gladkovskii, Jul 21 2012: (Start) Let E(x) be the e.g.f., then E(x) = 1/G(0) where G(k)= 1 - x/((1+x)*(2*k+1) - x*(1+x)*(2*k+1)/(x - (1+x)*(2*k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step). E(x) = 1 + x/(G(0)-x) where G(k)= 1 + 2*x + (1+x)*k - x*(1+x)*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). E(x) = G(0) where G(k)= 1 + x/((1+x)*(2*k+1) - x*(1+x)*(2*k+1)/(x + 2*(1+x)*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). (End) E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + 1/(k+1)/(1+x)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 27 2013 E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x/(x + (k+1)*(1+x)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 31 2013 a(n) = sum(k=0..n, (-1)^(n-k)*L(n,k)); L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014 a(n) = hypergeom([-n+1,-n],[],-1). - Peter Luschny, Apr 08 2015 D-finite with recurrence a(n) +(2*n-3)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Jul 20 2017 MATHEMATICA nn=30; CoefficientList[Series[Exp[x/(1+x)], {x, 0, nn}], x] Range[0, nn]! (* Harvey P. Dale, Jul 21 2011 *) PROG (Sage) A111884 = lambda n: hypergeometric([-n+1, -n], [], -1) [Integer(A111884(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 23 2014 CROSSREFS Unsigned row sums of A111596: A000262. Sequence in context: A125356 A293116 A066668 * A126514 A168025 A160431 Adjacent sequences: A111881 A111882 A111883 * A111885 A111886 A111887 KEYWORD sign,easy AUTHOR Wolfdieter Lang, Aug 23 2005 STATUS approved

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