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

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!)

Sequence in context: A030816 A030964 A030867 this_sequence A042545 A082936 A030935

Adjacent sequences: A007677 A007678 A007679 this_sequence A007681 A007682 A007683

nonn,easy

njas