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%I M4896 N2098 #229 Apr 30 2021 01:35:01
%S 0,1,1,-1,-1,13,-47,-73,2447,-16811,-15551,1726511,-18994849,10979677,
%T 2983409137,-48421103257,135002366063,10125320047141,-232033147779359,
%U 1305952009204319,58740282660173759,-1862057132555380307,16905219421196907793,527257187244811805207
%N Coefficients of Airey's converging factor.
%C A051711 times the coefficient in expansion of W(exp(x)) about x=1, where W is the Lambert function. - _Paolo Bonzini_, Jun 22 2016
%C The polynomials with coefficients in triangle A008517, evaluated at -1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001662/b001662.txt">Table of n, a(n) for n = 0..200</a> (terms 29 onwards updated by Sean A. Irvine, April 25 2019)
%H J. R. Airey, <a href="https://doi.org/10.1080/14786443708565133">The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions</a>, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].
%H J. A. Airey, <a href="/A001662/a001662.pdf"> The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions</a> [Annotated scanned copy]
%H Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H J. M. Borwein and R. M. Corless, <a href="http://docserver.carma.newcastle.edu.au/203/">Emerging tools for experimental mathematics</a>, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
%H R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, <a href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf">On the Lambert W Function</a>, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
%H R. M. Corless, D. J. Jeffrey and D. E. Knuth, <a href="http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/CorlessJeffreyKnuth.ps">A sequence of series for the Lambert W Function</a> (section 2.2).
%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions</a>, arXiv:math/0501052 [math.CA], 2005.
%H Vaclav Kotesovec, <a href="/A001662/a001662.jpg">Graph - the asymptotic ratio</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.
%H J. C. P. Miller, <a href="http://dx.doi.org/10.1017/S0305004100027602">A method for the determination of converging factors ...</a>, Proc. Camb. Phil. Soc., 48 (1952), 243-254.
%H J. C. P. Miller, <a href="/A001662/a001662_3.pdf">A method for the determination of converging factors ...</a> [Annotated scanned copy]
%H F. D. Murnaghan, <a href="https://doi.org/10.1073/pnas.69.2.440">Airey's converging factor</a>, Proc. Nat. Acad. Sci. USA, 69 (1972), 440-441.
%H F. D. Murnaghan and J. W. Wrench, Jr., <a href="https://catalog.hathitrust.org/Record/000618832">The Converging Factor for the Exponential Integral</a>, Report 1535, David Taylor Model Basin, U.S. Dept. of Navy, 1963 [ gives first 67 terms ].
%H N. J. A. Sloane, <a href="/A001662/a001662_2.pdf">Letter to F. D. Murnaghan, Apr 17, 1974</a>
%H J. W. Wrench, Jr., <a href="/A001662/a001662_1.pdf">Letter to N. J. A. Sloane, 24 Apr 1974</a>
%H P. Wynn, <a href="https://core.ac.uk/download/pdf/82359133.pdf">Converging factors for the Weber parabolic cylinder functions of complex argument I A</a>, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-736.
%H P. Wynn, <a href="https://core.ac.uk/download/pdf/82359133.pdf">Converging factors for the Weber parabolic cylinder functions of complex argument I B</a>, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 737-754.
%H P. Wynn, <a href="/A001662/a001662_4.pdf">Converging factors for the Weber parabolic cylinder functions ...</a> [Annotated scan of part 2 only]
%F 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]
%F 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
%F 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
%F From _Sergei N. Gladkovskii_, Nov 24 2012, Aug 22 2013: (Start)
%F Continued fractions:
%F G.f.: 2*x - x/G(0) where G(k) = 1 - 2*x*k + x*(k+1)/G(k+1).
%F G.f.: 2*x - 2*x/U(0) where U(k) = 1 + 1/(1 - 4*x*(k+1)/U(k+1)).
%F G.f.: A(x) = x/G(0) where G(k) = 1 - 2*x*(k+1) + x*(k+1)/G(k+1).
%F 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)
%F a(n) = 4^n * Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1). - _Paolo Bonzini_, Jun 23 2016
%F E.g.f.: 1/2*(LambertW(exp(4*x+1))-1). - _Vladimir Kruchinin_, Feb 18 2018
%F 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
%e 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
%p with(combinat); A001662 := proc(n) add((-1)^k*eulerian2(n-1,k),k=0..n-1) end:
%p seq(A001662(i),i=0..23); # _Peter Luschny_, Nov 13 2012
%t 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_ *)
%t 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 *)
%t len := 12; gf := (1/2) (LambertW[Exp[x + 1]] - 1);
%t ser := Series[gf, { x, 0, len}]; norm := Table[n! 4^n, {n, 0, len}];
%t CoefficientList[ser, x] * norm (* _Peter Luschny_, Jun 24 2019 *)
%o (Sage)
%o @CachedFunction
%o def eulerian2(n, k):
%o if k==0: return 1
%o elif k==n: return 0
%o return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
%o def A001662(n): return add((-1)^k*eulerian2(n-1,k) for k in (0..n-1))
%o [A001662(m) for m in (0..23)] # _Peter Luschny_, Nov 13 2012
%o (Maxima)
%o 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
%Y Cf. A051711, A032188, A274447, A274448, A340556.
%K sign,easy,nice
%O 0,6
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 07 1999
%E Reverted to converging factors definition by _Paolo Bonzini_, Jun 23 2016
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