A001811 Coefficients of Laguerre polynomials. (Formerly M5185 N2253)

1, 25, 450, 7350, 117600, 1905120, 31752000, 548856000, 9879408000, 185513328000, 3636061228800, 74373979680000, 1586644899840000, 35272336619520000, 816302647480320000, 19645683716026368000, 491142092900659200000, 12740803704070041600000

Offset

4,2

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.

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 = 4..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)

Index entries for sequences related to Laguerre polynomials.

Formula

a(n) = n!*n*(n-1)(n-2)(n-3)/(4!)^2. a(4)=1, a(n+1) = a(n) * (n+1)^2 / (n-3).

a(n) = A021009(n, 4), n >= 4.

E.g.f.: x^4/(4!*(1-x)^5).

If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^n*f(n,4,-5), (n >= 4). - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=4} 1/a(n) = 64*(Ei(1) - gamma - e) + 272/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=4} (-1)^n/a(n) = 544*(gamma - Ei(-1)) - 320/e - 944/3, where Ei(-1) = -A099285. (End)

Example

G.f. = x^4 + 25*x^5 + 450*x^6 + 7350*x^7 + 117600*x^8 + 1905120*x^9 + ...

Maple

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card<r), U=Sequence(Z, card>=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..19) ; # Zerinvary Lajos, Feb 07 2008

Mathematica

Table[n! n (n - 1) (n - 2) (n - 3)/(4!)^2, {n, 4, 20}] (* T. D. Noe, Aug 10 2012 *)

Prog

(Sage) [factorial(m) * binomial(m, 4) / 24 for m in range(4, 19)] # Zerinvary Lajos, Jul 05 2008

Crossrefs

Cf. A053495.

Cf. A001113, A001620, A091725, A099285.

Sequence in context: A260854 A001714 A016633 * A131279 A056069 A089386

Adjacent sequences: A001808 A001809 A001810 * A001812 A001813 A001814

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 07 2001

Corrected by T. D. Noe, Aug 10 2012

Status

approved