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 A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials. 8
 1, -2, -1, -4, -19, -110, -745, -5752, -49775, -476994, -5016069, -57462828, -712732987, -9521244982, -136356161873, -2084860795232, -33907076207495, -584602069590058, -10652917092110429, -204604743619641620, -4131502481607654739, -87507494737954740126 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..446 L. Comtet, Sur les coefficients de l'inverse de la sÃ©rie formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572. L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy) R. K. Guy, Letter to N. J. A. Sloane, Mar 1974 FORMULA From Vaclav Kotesovec, Aug 12 2015: (Start) G.f.: (1/Sum(k! x^k))^2. Expansion of (1-g(x))^2, where g(x) is the g.f. of A003319. a(n) ~ -2*n! * (1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - 31291/n^7 - 395360/n^8 - 5544754/n^9 - 85427259/n^10), for coefficients see A261214. For n>0, a(n) = A059439(n) - 2*A003319(n). For n>0, a(n) = Sum_{k=1..n} A260503(k) * Stirling1(n-1, k-1). (End) EXAMPLE A003319(n) / n! ~ 1 - 2/n - 1/(n*(n-1)) - 4/(n*(n-1)*(n-2)) - 19/(n*(n-1)*(n-2)*(n-3)) - 110/(n*(n-1)*(n-2)*(n-3)*(n-4)) - 745/(n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)) -  ... [coefficients are A259472] A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ... [coefficients are A260503] MATHEMATICA CoefficientList[Series[1/Sum[k! * x^k, {k, 0, 20}]^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 03 2015 *) CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x) * x^2 / ExpIntegralEi[1/x]^2, {x, 0, 25}]], x] (* Vaclav Kotesovec, Aug 03 2015 *) CROSSREFS Cf. A003319, A260503, A261214, A261239, A261253, A261254, A059439. Sequence in context: A013162 A010252 A032105 * A053565 A116603 A158356 Adjacent sequences:  A259469 A259470 A259471 * A259473 A259474 A259475 KEYWORD sign AUTHOR N. J. A. Sloane, Jul 03 2015, following a suggestion from R. K. Guy, Apr 29 1974 EXTENSIONS More terms from Vaclav Kotesovec, Aug 01 2015 New name from Vaclav Kotesovec, Aug 12 2015 Entry revised by Vaclav Kotesovec, Aug 12 2015 STATUS approved

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Last modified September 30 20:19 EDT 2020. Contains 337440 sequences. (Running on oeis4.)