Search: id:A007680
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%I A007680 M2861
%S A007680 1,3,10,42,216,1320,9360,75600,685440,6894720,76204800,918086400,
%T A007680 11975040000,168129561600,2528170444800,40537905408000,690452066304000,
%U A007680 12449059983360000,236887827111936000,4744158915944448000,99748982335242240000
%N A007680 (2n+1)*n!.
%C A007680 Denominators in series for sqrt(pi/4)*erf(x): sqrt(pi/4)*erf(x)= x/1 
               - x^3/3 + x^5/10 - x^7/42 + x^9/216 -+ ... This series is famous 
               for its bad convergence if x > 1
%C A007680 Appears to be the BinomialMean transform of A000354 (after truncating 
               the first term of A000354). (See A075271 for the definition of BinomialMean.) 
               - John W. Layman (layman(AT)math.vt.edu), Apr 16 2003
%C A007680 Number of permutations p of {1,2,...,n+2} such that max|p(i)-i|=n+1. 
               Example: a(1)=3 since only the permutations 312,231 and 321 of {1,
               2,3} satisfy the given condition. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jun 04 2003
%C A007680 Stirling transform of A000670(n+1)=[3,13,75,541,...] is a(n)=[3,10,42,
               216,...]. - Michael Somos Mar 04 2004
%C A007680 Stirling transform of a(n)=[2,10,42,216,...] is A052875(n+1)=[2,12,74,
               ...]. - Michael Somos Mar 04 2004
%C A007680 A related sequence also arises in evaluating indefinite integrals of 
               sec(x)^(2k+1), k=0, 1, 2, ... Letting u = sec(x) and d = sqrt(u^2-1), 
               one obtains a(0) = ln(u+d) 2*k*a(k) = (2*k-1)*u^(2*k-1)*d + a(k-1). 
               Viewing these as polynomials in u gives 2^k*k!*a(k) = a(0) + d*Sum(i=0..k-1){ 
               (2*i+1)*i!*2^i*u^(2*i+1) }, which is easily proved by induction. 
               Apart from the power of 2, which could be incorporated into the definition 
               of u (or by looking at erf(ix/2)/ i (i=sqrt(-1)), the sum's coefficients 
               form our series and are the reciprocals of the power series terms 
               for -sqrt(-pi/4)*erf(ix/2)). This yields a direct but somewhat mysterious 
               relationship between the power series of erf(x) and integrals involving 
               sec(x). - William A. Huber (whuber(AT)quantdec.com), Mar 14 2002
%D A007680 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007680 E. Deutsch, Math. Magazine, vol. 74, No. 5, 2001, p. 404, problem Q915.
%D A007680 H. W. Gould, A class of binomial sums and a series transformation, Utilitas 
               Math., 45 (1994), 71-83.
%D A007680 N. Wirth, Systematisches Programmieren, 1975, exercise 9.3
%D A007680 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the 
               Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.1.1.
%H A007680 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Erf.html">Erf</a>
%H A007680 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/VOL9/Spivey/spivey7.pdf">The k-Binomial Transforms and the Hankel 
               Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%F A007680 E.g.f.: (1+x)/(1-x)^2.
%F A007680 This is the binomial mean transform of A000354 (after truncating the 
               first term). See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), 
               Feb 26 2006
%t A007680 Table[(2n + 1)*n!, {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 08 2006
%o A007680 (PARI) a(n)=if(n<0,0,(2*n+1)*n!)
%Y A007680 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%Y A007680 Appears in A167546.
%Y A007680 Equals the rows sums of A167556.
%Y A007680 (End)
%Y A007680 Sequence in context: A030816 A030964 A030867 this_sequence A143523 A042545 
               A151084
%Y A007680 Adjacent sequences: A007677 A007678 A007679 this_sequence A007681 A007682 
               A007683
%K A007680 nonn,easy
%O A007680 0,2
%A A007680 N. J. A. Sloane (njas(AT)research.att.com).

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