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A007680 (2n+1)*n!.
(Formerly M2861)
18
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) = 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

REFERENCES

H. W. Gould, A class of binomial sums and a series transformation, 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.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

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

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

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!)

(MAGMA)[(2*n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 20 2011

CROSSREFS

From Johannes W. Meijer, Nov 12 2009: (Start)

Appears in A167546.

Equals the rows sums of A167556.

(End)

Sequence in context: A030964 A030867 A186360 * A232606 A185621 A190657

Adjacent sequences:  A007677 A007678 A007679 * A007681 A007682 A007683

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 23 12:36 EDT 2014. Contains 247167 sequences.