A001811 Coefficients of Laguerre polynomials.

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

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

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

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.

Crossrefs

Cf. A053495.

Cf. A001113, A001620, A091725, A099285.

Programs

Maple

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, 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 *)

Sage

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

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)

Extensions

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

Corrected by T. D. Noe, Aug 10 2012