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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

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

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.

Crossrefs

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

Negative coefficient of x of polynomials in A098503.

Programs

Magma

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

Mathematica

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

PARI

a(n)=(2*n+6)!/(n+3)!/2^(n+3)/3-(2*n+4)!/(n+2)!/2^(n+2)

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