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 A007680 a(n) = (2n+1)*n!. (Formerly M2861) 30
 1, 3, 10, 42, 216, 1320, 9360, 75600, 685440, 6894720, 76204800, 918086400, 11975040000, 168129561600, 2528170444800, 40537905408000, 690452066304000, 12449059983360000, 236887827111936000, 4744158915944448000, 99748982335242240000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 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 Stirling transform of A000670(n+1) = [3, 13, 75, 541, ...] is a(n) = [3, 10, 42, 216, ...]. - Michael Somos, Mar 04 2004 Stirling transform of a(n) = [2, 10, 42, 216, ...] is A052875(n+1) = [2, 12, 74, ...]. - Michael Somos, Mar 04 2004 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 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 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 REFERENCES H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). N. Wirth, Systematisches Programmieren, 1975, exercise 9.3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..400 Emeric Deutsch, Problem Q915, Math. Magazine, vol. 74, No. 5, 2001, p. 404. H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy) Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015. M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1. Eric Weisstein's World of Mathematics, Erf Wikipedia, Factorial base FORMULA E.g.f.: (1+x)/(1-x)^2. 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 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 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 a(n-2) = (A208528(n)+A208529(n))/2, for n>=2. - Luis Manuel Rivera Martínez, Mar 05 2014 D-finite with recurrence: (-2*n+1)*a(n) +n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 27 2020 EXAMPLE 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 MAPLE [(2*n+1)*factorial(n)\$n=0..20]; # Muniru A Asiru, Jan 01 2019 MATHEMATICA Table[(2n + 1)*n!, {n, 0, 20}] (* Stefan Steinerberger, Apr 08 2006 *) PROG (PARI) {a(n) = if( n<0, 0, (2*n+1) * n!)}; /* Michael Somos, Mar 04 2004 */ (MAGMA)[(2*n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 20 2011 (GAP) a:=List([0..20], n->(2*n+1)*Factorial(n));; Print(a); # Muniru A Asiru, Jan 01 2019 CROSSREFS From Johannes W. Meijer, Nov 12 2009: (Start) Appears in A167546. Equals the rows sums of A167556. (End) Sequence in context: A263823 A030867 A186360 * A232606 A334270 A185621 Adjacent sequences:  A007677 A007678 A007679 * A007681 A007682 A007683 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified July 3 09:17 EDT 2020. Contains 335417 sequences. (Running on oeis4.)