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 A006252 Expansion of e.g.f. 1/(1 - log(1+x)). (Formerly M1275) 29
 1, 1, 1, 2, 4, 14, 38, 216, 600, 6240, 9552, 319296, -519312, 28108560, -176474352, 3998454144, -43985078784, 837126163584, -12437000028288, 237195036797184, -4235955315745536, 85886259443020800, -1746536474655406080, 38320721602434017280, -864056965711935974400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS From Michael Somos, Mar 04 2004: (Start) Stirling transform of a(n+1)=[1,1,2,4,14,38,...] is A000255(n)=[1,3,11,53,309,...]. Stirling transform of 2*a(n)=[2,2,4,8,28,...] is A052849(n)=[2,4,12,48,240,...]. Stirling transform of a(n)=[1,1,2,4,14,38,216,...] is A000142(n)=[1,2,6,24,120,...]. Stirling transform of a(n-1)=[1,1,1,2,4,14,38,...] is A000522(n-1)=[1,2,5,16,65,...]. Stirling transform of a(n-1)=[0,1,1,2,4,14,38,...] is A007526(n-1)=[0,1,4,15,64,...]. (End) For n > 0: a(n) = sum of n-th row in triangle A048594. - Reinhard Zumkeller, Mar 02 2014 Coefficients in a factorial series representation of the exponential integral: exp(z)*E_1(z) = Sum_{n >= 0} (-1)^n*a(n)/(z)_n, where (z)_n denotes the rising factorial z*(z + 1)*...*(z + n) and E_1(z) = Integrate_{t = z..inf} exp(-t)/t dt. See Weninger, equation 6.4. - Peter Bala, Feb 12 2019 REFERENCES G. Pólya, Induction and Analogy in Mathematics. Princeton Univ. Press, 1954, p. 9. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..400 Beáta Bényi, Daniel Yaqubi, Mixed coloured permutations, arXiv:1903.07450 [math.CO], 2019. Takao Komatsu, Amalia Pizarro-Madariaga, Harmonic numbers associated with inversion numbers in terms of determinants, Turkish Journal of Mathematics (2019) Vol. 43, 340-354. E. J. Weniger, Summation of divergent power series by means of factorial series arXiv:1005.0466v1 [math.NA], 2010. FORMULA a(n) = Sum_{k=0..n} k!*stirling1(n, k). - Vladeta Jovovic, Sep 08 2002 a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(-x)*d/dx. Row sums of A048594. Cf. A007840. - Peter Bala, Nov 25 2011 E.g.f.: 1/(1-log(1+x)) = 1 + x/(1-x + x/(2-x + 4*x/(3-2*x + 9*x/(4-3*x + 16*x/(5-4*x + 25*x/(6-5*x +...)))))), a continued fraction. - Paul D. Hanna, Dec 31 2011 a(n)/n! ~ -(-1)^n / (n * (log(n))^2) * (1 - 2*(1 + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2018 MATHEMATICA With[{nn=30}, CoefficientList[Series[1/(1-Log[1+x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 12 2016 *) PROG (PARI) a(n)=if(n<0, 0, n!*polcoeff(1/(1-log(1+x+x*O(x^n))), n)) (PARI) {a(n)=local(CF=1+x*O(x^n)); for(k=0, n-1, CF=1/((n-k+1)-(n-k)*x+(n-k+1)^2*x*CF)); n!*polcoeff(1+x/(1-x+x*CF), n, x)} /* Paul D. Hanna, Dec 31 2011 */ (Haskell) a006252 0 = 1 a006252 n = sum \$ a048594_row n  -- Reinhard Zumkeller, Mar 02 2014 (Sage) def A006252_list(len):     f, R, C = 1, [1], [1]+[0]*len     for n in (1..len):         f *= n         for k in range(n, 0, -1):             C[k] = -C[k-1]*((k-1)/(k) if k>1 else 1)         C[0] = -sum(C[k] for k in (1..n))         R.append(C[0]*f)     return R print(A006252_list(24)) # Peter Luschny, Feb 21 2016 CROSSREFS Sequence in context: A053623 A035010 A055540 * A079995 A279322 A152011 Adjacent sequences:  A006249 A006250 A006251 * A006253 A006254 A006255 KEYWORD sign AUTHOR STATUS approved

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Last modified October 23 23:52 EDT 2020. Contains 337975 sequences. (Running on oeis4.)