A001812 Coefficients of Laguerre polynomials. (Formerly M5257 N2289)

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

Offset

5,2

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 = 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.

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)

Example

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

Mathematica

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

Prog

(Sage) [factorial(m) * binomial(m, 5) / 120 for m in range(5, 23)] # Zerinvary Lajos, Jul 05 2008
(PARI) for(n=5, 20, print1(((n!/5!)^2)/(n-5)!, ", ")) \\ G. C. Greubel, May 11 2018
(Magma) [((Factorial(n)/Factorial(5))^2)/Factorial(n-5): n in [5..20]]; // G. C. Greubel, May 11 2018

Crossrefs

Cf. A001113, A001620, A021009, A091725, A099285.

Sequence in context: A243569 A145150 A166790 * A229680 A169836 A335220

Adjacent sequences: A001809 A001810 A001811 * A001813 A001814 A001815

Keyword

nonn

Author

N. J. A. Sloane

Status

approved