A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials.

1, -2, -1, -4, -19, -110, -745, -5752, -49775, -476994, -5016069, -57462828, -712732987, -9521244982, -136356161873, -2084860795232, -33907076207495, -584602069590058, -10652917092110429, -204604743619641620, -4131502481607654739, -87507494737954740126

Offset: 0

Author

N. J. A. Sloane, Jul 03 2015, following a suggestion from R. K. Guy, Apr 29 1974

Keywords

• sign

Examples

			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]
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..446
• L. Comtet, Sur les coefficients de l'inverse de la série 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

Crossrefs

Cf. A003319, A260503, A261214, A261239, A261253, A261254, A059439.

Programs

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 *)

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)

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