A001812 Coefficients of Laguerre polynomials.

1, 36, 882, 18816, 381024, 7620480, 153679680, 3161410560, 66784798080, 1454424491520, 32724551059200, 761589551923200, 18341615042150400, 457129482588979200, 11787410229615820800, 314330939456421888000, 8663746518767628288000, 246661959710796005376000

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = x^5 + 36*x^6 + 882*x^7 + 18816*x^8 + 381024*x^9 + 7620480*x^10 + ...
		

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 = 5..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. A001113, A001620, A021009, A091725, A099285.

Programs

Magma

[((Factorial(n)/Factorial(5))^2)/Factorial(n-5): n in [5..20]]; // G. C. Greubel, May 11 2018

Mathematica

Table[((n!/5!)^2)/(n-5)!, {n, 5, 20}] (* T. D. Noe, Aug 10 2012 *)

PARI

for(n=5,20, print1(((n!/5!)^2)/(n-5)!, ", ")) \\ G. C. Greubel, May 11 2018

Sage

[factorial(m) * binomial(m, 5) / 120 for m in range(5,23)] # Zerinvary Lajos, Jul 05 2008

Formula

a(n) = (-1)*A021009(n, 5), n >= 5.

a(n) = ((n!/5!)^2)/(n-5)!, n >= 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-1)*f(n,5,-6), (n>=5). - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=5} 1/a(n) = 375*(gamma - Ei(1)) + 150*e + 175/2, where e = A001113, gamma = A001620, and Ei(1) = A091725.

Sum_{n>=5} (-1)^(n+1)/a(n) = 5225*(gamma - Ei(-1)) - 3100/e - 18125/6, where Ei(-1) = -A099285. (End)