A274448 -id:A274448 - cp's OEIS frontend

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

N. J. A. Sloane

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.

			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

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).

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]

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

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

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 *)

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 */

@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

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

More terms from James Sellers, Dec 07 1999

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

A274447 Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.

Original entry on oeis.org

1, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 778870772857, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793, 527257187244811805207

Offset: 0

Paolo Bonzini, Jun 23 2016

a(17) is the first term that differs from A001662.

			W(exp(x)) = 1 + (x-1)/2 + (x-1)^2/16 - (x-1)^3/192 - ...

Alois P. Heinz, Table of n, a(n) for n = 0..446 (first 151 terms from G. C. Greubel)
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).

Cf. A001662, A051711, A274448.

a:= n-> numer(coeftayl(LambertW(exp(x)), x=1, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 08 2012
# For large n much faster is:
q := proc(n) if n=0 then 1 else add((-1)^k*combinat[eulerian2](n-1, k), k=0..n-1) fi end: A001662 := n -> numer(q(n)/n!):
seq(A001662(n), n=0..100):  # Peter Luschny, Nov 13 2012

CoefficientList[ Series[ ProductLog[ Exp[1+x] ], {x, 0, 22}], x] // Numerator (* Jean-François Alcover, Oct 15 2012 *)
a[0] = 1; a[n_] := 1/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}] // Numerator; Array[a, 30, 0] (* Jean-François Alcover, Feb 13 2016, after Vladimir Kruchinin *)

a(n):=num(if n=0 then 1 else 1/n!*(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 */

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    if k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
def q(n):
    return add((-1)^k*eulerian2(n-1, k) for k in (0..n-1)) if n>0 else 1
A001662 = lambda n: (q(n)/factorial(n)).numerator()
[A001662(n) for n in (0..22)]  # Peter Luschny, Nov 13 2012

a(n) = A001662(n)/gcd(A001662(n),A051711(n)).
From Vladimir Kruchinin, Nov 11 2012: (Start)
a(n) = numerator(1/n!*(Sum_{u=2..n} stirling2(n,u)*(Sum_{k=1..u-1} ((u+k-1)!*Sum_{j=1..k} 2^(-u-j)/(k-j)!*Sum_{l=1..j} (-1)^(l)/((j-l)!)*Sum_{i=0..l} (l^(u+j-i-1))/((l-i)!*i!*(u+j-i-1)!)))+1/2)).
a(n) = numerator((1/n!)*Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^j/(k-j)!)*2^(n-j-1)*Sum_{i=0..j} (1/i!)*Stirling1(n-i+j-1,j-i)/(n-i+j-1)!), n>0, a(0)=1. (End)
a(n) = numerator(q(n)/n!) where q(n) = add_{k=0..n-1}(-1)^k*E2(n-1,k) if n>0 and 1 otherwise, E2 the second-order Eulerian numbers. - Peter Luschny, Nov 13 2012
a(n) := numerator(1/n!*Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1)). - Paolo Bonzini, Jun 23 2016

cp's OEIS Frontend

A001662 Coefficients of Airey's converging factor.

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