|
|
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
|
L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
|
|
FORMULA
|
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) = 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
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|