login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001662 Coefficients of Airey's converging factor.
(Formerly M4896 N2098)
9
0, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 10125320047141, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793, 527257187244811805207 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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.

REFERENCES

J. R. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other function, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].

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, 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).

P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-754 (two parts).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

J. A. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other function [Annotated scanned copy]

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 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.

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.

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 ... [Annotated scan of part 2 only]

FORMULA

The g.f. A(x) satisfies exp(A(x)) = 1 + 2*x - A(x).

From Vladimir Kruchinin, Nov 11 2012: (Start)

E.g.f. A(x), B(x) = A(x)-1 satisfies the differential equation B'(x) = 1/(1+B(x)).

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.

(End)

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

G.f.: 2*x - x/G(0) where G(k) = 1 - 2*x*k + x*(k+1)/G(k+1); (continued fraction, 1-step).

G.f.: 2*x - 2*x/U(0) where U(k) = 1 + 1/(1 - 4*x*(k+1)/U(k+1)); (continued fraction, 2-step).

G.f.: A(x) = x/G(0) ; G(k) = 1 - 2*x*(k+1) + x*(k+1)/G(k+1); (continued fraction). (End)

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) )) ); (continued fraction). - Sergei N. Gladkovskii, Aug 22 2013

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

MAPLE

with(combinat); A001662 := proc(n) add((-1)^k*eulerian2(n-1, k), k=0..n-1) end:

seq(A001662(i), i=0..23);

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-Fran├žois Alcover, Jul 26 2013, after Vladimir Kruchinin *)

PROG

(Sage)

@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)]

(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

CROSSREFS

Cf. A051711, A032188, A274447, A274448.

Sequence in context: A127305 A207992 A274447 * A031390 A113943 A222962

Adjacent sequences:  A001659 A001660 A001661 * A001663 A001664 A001665

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Dec 07 1999

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 19 02:01 EST 2018. Contains 317332 sequences. (Running on oeis4.)