A001688 4th forward differences of factorial numbers A000142.

9, 53, 362, 2790, 24024, 229080, 2399760, 27422640, 339696000, 4536362880, 64988179200, 994447238400, 16190733081600, 279499828608000, 5100017213491200, 98087346669312000, 1983334021853184000, 42063950934061056000, 933754193111900160000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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 = 0..100
• A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A001563, A001564, A001565, A001689, A095177.

Programs

Mathematica

Table[(n^4 + 6*n^3 + 17*n^2 + 20*n + 9) n!, {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)
Differences[Range[0,30]!,4] (* Harvey P. Dale, Jun 06 2017 *)

PARI

a(n)=if(n<0,0,n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9))

Formula

For n>=0 a(n) = n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9). - Benoit Cloitre, Jun 10 2004

G.f.: -log(-x+1)+1+2/(x-1)^4*x*(4-3*x+2*x^2). - Simon Plouffe, Master's Thesis, Uqam 1992

E.g.f.: (9 + 8*x + 6*x^2 + x^4)/(1 - x)^5. - Ilya Gutkovskiy, Jan 20 2017

a(n) = (n+5)*a(n-1) - (n-1)*a(n-2) with a(0) = 9 and a(1) = 53. Cf. A095177. - Peter Bala, Jul 22 2021