A000906 Exponential generating function: 2*(1+3*x)/(1-2*x)^(7/2). (Formerly M2124 N0841)

2, 20, 210, 2520, 34650, 540540, 9459450, 183783600, 3928374450, 91662070500, 2319050383650, 63246828645000, 1849969737866250, 57775977967207500, 1918987839625106250, 67548371954803740000, 2511955082069264081250

Offset

0,1

Comments

Ramanujan polynomials -psi_{n+2}(n+2,x) evaluated at 1.

With offset 2, second Eulerian transform of 0,1,2,3,4... - Ross La Haye, Mar 05 2005

With offset 1, a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 30 2016

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.

F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 152.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.

C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.

Formula

a(n) = (2n+5)!!/3 - (2n+3)!!.

a(n) -2*(n+4)*a(n-1) +3*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Feb 20 2013

a(n) ~ 2^(n+7/2)*n^(n+3)/(3*exp(n)). - Ilya Gutkovskiy, Aug 17 2016

a(n) = (2n+3)!/( 3!*n!*2^(n-1) ). - G. C. Greubel, May 15 2018

Example

G.f. = 2 + 20*x + 210*x^2 + 2520*x^3 + 34650*x^4 + 540540*x^5 + ...

Mathematica

Table[(2 n + 5)!!/3 - (2 n + 3)!!, {n, 0, 20}] (* Vincenzo Librandi, Apr 11 2012 *)

Prog

(PARI) a(n)=(2*n+6)!/(n+3)!/2^(n+3)/3-(2*n+4)!/(n+2)!/2^(n+2)
(Magma) [Factorial(2*n+3)/(6*Factorial(n)*2^(n-1)): n in [0..30]]; // G. C. Greubel, May 15 2018

Crossrefs

a(n) = 2*A000457(n) = A051577(n+1) - A001147(n+2).

Negative coefficient of x of polynomials in A098503.

Sequence in context: A173499 A067636 A226301 * A308945 A356853 A199761

Adjacent sequences: A000903 A000904 A000905 * A000907 A000908 A000909

Keyword

nonn

Author

N. J. A. Sloane

Status

approved