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A007680 a(n) = (2n+1)*n!.

%I A007680 M2861 #111 Apr 02 2025 05:16:18
%S A007680 1,3,10,42,216,1320,9360,75600,685440,6894720,76204800,918086400,
%T A007680 11975040000,168129561600,2528170444800,40537905408000,
%U A007680 690452066304000,12449059983360000,236887827111936000,4744158915944448000,99748982335242240000
%N A007680 a(n) = (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 -+ ...
%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_, 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_, 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) = log(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
%C A007680 When written in factoradic ("factorial base"), this sequence from a(1) onwards gives the smallest number containing two adjacent digits, increasing when read from left to right, whose difference is n-1. - _Christian Perfect_, May 03 2016
%C A007680 a(n-1)^2 is the number of permutations p of [1..2n] such that Sum_{i=1..2n} abs(p(i)-i) = 2n^2-2. - _Fang Lixing_, Dec 07 2018
%C A007680 A standard series for the calculation of coordinates on a clothoid (also called cornuspiral):
%C A007680   x = s*(a(0) - (tau^2/a(2)) + (tau^4/a(4)) - (tau^6/a(6)) + ...)
%C A007680   y = s*((tau/a(1)) + (tau^3/a(3)) - (tau^5/a(5)) + ...).
%C A007680   s is the arclength from the clothoids origin to the desired point p(x,y). The tangent at the clothoids origin intersects with the tangent at the point p(x,y) with an angle of tau. - _Thomas Scheuerle_, Oct 13 2021
%C A007680 a(n) = P_n(1) where P_n(x) is the Pidduck polynomials. - _Michael Somos_, May 27 2023
%D A007680 H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
%D A007680 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007680 N. Wirth, Systematisches Programmieren, 1975, exercise 9.3
%H A007680 Vincenzo Librandi, <a href="/A007680/b007680.txt">Table of n, a(n) for n = 0..400</a>
%H A007680 Emeric Deutsch, <a href="http://www.jstor.org/stable/2691040">Problem Q915</a>, Math. Magazine, vol. 74, No. 5, 2001, p. 404.
%H A007680 H. W. Gould, <a href="/A007680/a007680.pdf">A class of binomial sums and a series transform</a>, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
%H A007680 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H A007680 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%H A007680 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erf.html">Erf</a>
%H A007680 Wikipedia, <a href="http://en.wikipedia.org/wiki/Factoradic">Factorial base</a>
%H A007680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pidduck_polynomials">Pidduck polynomials</a>
%H A007680 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 5.
%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
%F A007680 E.g.f.: (of aerated sequence) 1+x^2/2+sqrt(pi)*(x+x^3/4)*exp(x^2/4)*ERF(x/2). - _Paul Barry_, Apr 11 2010
%F A007680 G.f.: 1 + x*G(0), where G(k)= 1 + x*(k+1)/(1 - (k+2)/(k+2 + (k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 08 2013
%F A007680 a(n-2) = (A208528(n)+A208529(n))/2, for n>=2. - _Luis Manuel Rivera Martínez_, Mar 05 2014
%F A007680 D-finite with recurrence: (-2*n+1)*a(n) +n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 27 2020
%F A007680 Sum_{n>=0} 1/a(n) = sqrt(Pi)*erfi(1)/2 = A019704 * A099288 = A347910. - _Amiram Eldar_, Oct 07 2020
%F A007680 Sum_{n>=0} (-1)^n/a(n) = A347909 . - _R. J. Mathar_, Sep 30 2021
%e A007680 G.f. = 1 + 3*x + 10*x^2 + 42*x^3 + 216*x^4 + 1320*x^5 + 9360*x^6 + ... - _Michael Somos_, Jan 01 2019
%p A007680 [(2*n+1)*factorial(n)$n=0..20]; # _Muniru A Asiru_, Jan 01 2019
%t A007680 Table[(2n + 1)*n!, {n, 0, 20}] (* _Stefan Steinerberger_, Apr 08 2006 *)
%o A007680 (PARI) {a(n) = if( n<0, 0, (2*n+1) * n!)}; /* _Michael Somos_, Mar 04 2004 */
%o A007680 (Magma)[(2*n+1)*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Aug 20 2011
%o A007680 (GAP) a:=List([0..20],n->(2*n+1)*Factorial(n));; Print(a); # _Muniru A Asiru_, Jan 01 2019
%Y A007680 From _Johannes W. Meijer_, Nov 12 2009: (Start)
%Y A007680 Appears in A167546.
%Y A007680 Equals the rows sums of A167556.
%Y A007680 (End)
%Y A007680 Cf. A019704, A099288.
%K A007680 nonn,easy
%O A007680 0,2
%A A007680 _N. J. A. Sloane_