A001823 Central factorial numbers: column 2 in triangle A008956.

0, 9, 259, 1974, 8778, 28743, 77077, 179452, 375972, 725781, 1312311, 2249170, 3686670, 5818995, 8892009, 13211704, 19153288, 27170913, 37808043, 51708462, 69627922, 92446431, 121181181, 157000116, 201236140, 255401965, 321205599, 400566474, 495632214

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
• 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 = 1..1000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Crossrefs

A bisection of A181888.

Column 2 in triangle A008956.

Programs

Maple

A001823:=-(9+196*z+350*z**2+84*z**3+z**4)/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

Table[1/90*n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n + 7), {n, 40}] (* Stefan Steinerberger, Apr 15 2006 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,9,259,1974,8778,28743,77077},30] (* Harvey P. Dale, Jun 09 2013 *)

Formula

a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n+7)/90.

If we replace n with n-1/2 in this formula we get 16*A000586(n).

G.f.: z*(9+196*z+350*z**2+84*z**3+z**4)/(1-z)^7.

a(1)=0, a(2)=9, a(3)=259, a(4)=1974, a(5)=8778, a(6)=28743, a(7)=77077, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 09 2013

Extensions

More terms from Stefan Steinerberger, Apr 15 2006