A062193 Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

1, 24, 420, 6720, 105840, 1693440, 27941760, 479001600, 8562153600, 159826867200, 3116623910400, 63465795993600, 1348648164864000, 29877743960064000, 689322235650048000, 16543733655601152000, 412559358036553728000, 10678006913887272960000, 286526518855975157760000

Offset

0,2

Links

Harry J. Smith, Table of n, a(n) for n = 0..100

Index entries for sequences related to Laguerre polynomials.

Formula

E.g.f.: (1+15*x+30*x^2+10*x^3)/(1-x)^9.

a(n) = A062139(n+3, 3).

a(n) = (n+3)!*binomial(n+5, 5)/3!.

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-3) = (-1)^(n-1)*f(n,3,-6), (n>=3). - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 75*(Ei(1) - gamma) - 30*e - 65/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=0} (-1)^n/a(n) = 315*(gamma - Ei(-1)) - 180/e - 735/4, where Ei(-1) = -A099285. (End)

Mathematica

With[{nn=20}, CoefficientList[Series[(1+15*x+30*x^2+10*x^3)/(1-x)^9, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Mar 02 2018 *)

Prog

(Sage) [binomial(n, 5)*factorial (n-2)/6 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009
(PARI) { f=2; for (n=0, 100, f*=n + 3; write("b062193.txt", n, " ", f*binomial(n + 5, 5)/6) ) } \\ Harry J. Smith, Aug 02 2009
(PARI) my(x='x+O('x^30)); Vec(serlaplace((1+15*x+30*x^2+10*x^3)/(1-x)^9)) \\ G. C. Greubel, May 11 2018
(Magma) [Factorial(n+3)*binomial(n+5, 5)/Factorial(3): n in [0..30]]; // G. C. Greubel, May 11 2018

Crossrefs

Cf. A001710, A005990, A005461, A062139.

Cf. A001113, A001620, A091725, A099285.

Sequence in context: A016326 A287994 A072975 * A016268 A175199 A051546

Adjacent sequences: A062190 A062191 A062192 * A062194 A062195 A062196

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 19 2001

Status

approved