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
KEYWORD
nonn
AUTHOR
STATUS
approved