A001708 Generalized Stirling numbers. (Formerly M5095 N2206)

1, 20, 295, 4025, 54649, 761166, 11028590, 167310220, 2664929476, 44601786944, 784146622896, 14469012689040, 279870212258064, 5667093514231200, 119958395537083104, 2650594302549806976, 61049697873641191296, 1463708634867162093312, 36482312832434713195776

Offset

0,2

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=2) ~ exp(-x)/x^5*(1 - 20/x + 295/x^2 - 4025/x^3 + 54649/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - Johannes W. Meijer, Oct 20 2009

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..100

D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

Formula

E.g.f.: ( log ( 1 - x ))^4 / 24 ( 1 - x )^2.

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+4, 4)*2^k*Stirling1(n+4, k+4). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a - j), then a(n-4) = |f(n,4,2)| for n >= 4. - Milan Janjic, Dec 21 2008

Mathematica

With[{nn=20}, Drop[CoefficientList[Series[Log[1-x]^4/(24(1-x)^2), {x, 0, nn}], x]Range[0, nn]!, 4]] (* Harvey P. Dale, Oct 24 2011 *)

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace((log(1-x))^4/(24*(1-x)^2))) \\ Michel Marcus, Feb 04 2022

Crossrefs

Sequence in context: A223956 A132168 A069326 * A016255 A322052 A250014

Adjacent sequences: A001705 A001706 A001707 * A001709 A001710 A001711

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

Status

approved