0,2

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 (layman(AT)math.vt.edu), 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 (deutsch(AT)duke.poly.edu), 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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. Deutsch, Math. Magazine, vol. 74, No. 5, 2001, p. 404, problem Q915.

H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.

N. Wirth, Systematisches Programmieren, 1975, exercise 9.3

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.

Eric Weisstein's World of Mathematics, Erf

M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

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

Table[(2n + 1)*n!, {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006

(PARI) a(n)=if(n<0, 0, (2*n+1)*n!)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)

Appears in A167546.

Equals the rows sums of A167556.

(End)

Sequence in context: A030816 A030964 A030867 this_sequence A143523 A042545 A151084

Adjacent sequences: A007677 A007678 A007679 this_sequence A007681 A007682 A007683

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).