A001662 Coefficients of Airey's converging factor.

0, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 10125320047141, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793, 527257187244811805207

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Comments

A051711 times the coefficient in expansion of W(exp(x)) about x=1, where W is the Lambert function. - Paolo Bonzini, Jun 22 2016

The polynomials with coefficients in triangle A008517, evaluated at -1.

Examples

			G.f. = x + x^2 - x^3 - x^4 + 13*x^5 - 47*x^6 - 73*x^7 + 2447*x^8 + ... - _Michael Somos_, Jun 23 2019
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms 29 onwards updated by Sean A. Irvine, April 25 2019)
• J. R. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].
• J. A. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions [Annotated scanned copy]
• Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
• R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
• R. M. Corless, D. J. Jeffrey and D. E. Knuth, A sequence of series for the Lambert W Function (section 2.2).
• D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
• Vaclav Kotesovec, Graph - the asymptotic ratio
• Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
• J. C. P. Miller, A method for the determination of converging factors ..., Proc. Camb. Phil. Soc., 48 (1952), 243-254.
• J. C. P. Miller, A method for the determination of converging factors ... [Annotated scanned copy]
• F. D. Murnaghan, Airey's converging factor, Proc. Nat. Acad. Sci. USA, 69 (1972), 440-441.
• F. D. Murnaghan and J. W. Wrench, Jr., The Converging Factor for the Exponential Integral, Report 1535, David Taylor Model Basin, U.S. Dept. of Navy, 1963 [ gives first 67 terms ].
• N. J. A. Sloane, Letter to F. D. Murnaghan, Apr 17, 1974
• J. W. Wrench, Jr., Letter to N. J. A. Sloane, 24 Apr 1974
• P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I A, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-736.
• P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I B, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 737-754.
• P. Wynn, Converging factors for the Weber parabolic cylinder functions ... [Annotated scan of part 2 only]

Crossrefs

Cf. A051711, A032188, A274447, A274448, A340556.

Programs

Maple

with(combinat); A001662 := proc(n) add((-1)^k*eulerian2(n-1,k),k=0..n-1) end:
seq(A001662(i),i=0..23); # Peter Luschny, Nov 13 2012

Mathematica

a[0] = 0; a[n_] := Sum[ (n+k-1)! * Sum[ (-1)^j/(k-j)! * Sum[ 1/i! * StirlingS1[n-i+j-1, j-i] / (n-i+j-1)!, {i, 0, j}] * 2^(n-j-1), {j, 0, k}], {k, 0, n-1}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 26 2013, after Vladimir Kruchinin *)
a[ n_] := If[ n < 1, 0, 2^(n - 1) Sum[ (-2)^-j StirlingS1[n - i + j - 1, j - i] Binomial[n + k - 1, n + j - 1] Binomial[n + j - 1, i], {k, 0, n - 1}, {j, 0, k}, {i, 0, j}]]; (* Michael Somos, Jun 23 2019 *)
len := 12; gf := (1/2) (LambertW[Exp[x + 1]] - 1);
ser := Series[gf, { x, 0, len}]; norm := Table[n! 4^n, {n, 0, len}];
CoefficientList[ser, x] * norm (* Peter Luschny, Jun 24 2019 *)

Maxima

a(n):= if n=0 then 1 else (sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum((1/i! *stirling1(n-i+j-1, j-i))/(n-i+j-1)!, i, 0, j))*2^(n-j-1), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Nov 11 2012 */

SageMath

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    elif k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
def A001662(n): return add((-1)^k*eulerian2(n-1,k) for k in (0..n-1))
[A001662(m) for m in (0..23)] # Peter Luschny, Nov 13 2012

Formula

Let b(n) = 0, 1, -1, 1, 1, -13,.. be the sequence with all signs but one reversed: b(1)=a(1), b(n)=-a(n) for n<>1. Define the e.g.f. B(x) = 2*Sum_{n>=0} b(n)*(x/2)^n/n!. B(x) satisfies exp(B(x)) = 1 + 2*x - B(x). [Bernstein/Sloane S52]

Similarly, c(0)=1, c(n)=-a(n+1) are the alternating row sums of the second-order Eulerian numbers A340556, or c(n) = E2poly(n,-1). - Peter Luschny, Feb 13 2021

a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^j/(k-j)!)*Sum_{i=0..j} ((1/i!)*Stirling1(n-i+j-1,j-i)/(n-i+j-1)!)*2^(n-j-1), n > 0, a(0)=1. - Vladimir Kruchinin, Nov 11 2012

From Sergei N. Gladkovskii, Nov 24 2012, Aug 22 2013: (Start)

Continued fractions:

G.f.: 2*x - x/G(0) where G(k) = 1 - 2*x*k + x*(k+1)/G(k+1).

G.f.: 2*x - 2*x/U(0) where U(k) = 1 + 1/(1 - 4*x*(k+1)/U(k+1)).

G.f.: A(x) = x/G(0) where G(k) = 1 - 2*x*(k+1) + x*(k+1)/G(k+1).

G.f.: 2*x - x*W(0) where W(k) = 1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/W(k+1)))). (End)

a(n) = 4^n * Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1). - Paolo Bonzini, Jun 23 2016

E.g.f.: 1/2*(LambertW(exp(4*x+1))-1). - Vladimir Kruchinin, Feb 18 2018

a(0) = 0; a(1) = 1; a(n) = 2 * a(n-1) - Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020

Extensions

More terms from James Sellers, Dec 07 1999

Reverted to converging factors definition by Paolo Bonzini, Jun 23 2016