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A001688 4th forward differences of factorial numbers A000142.

%I A001688 M4636 N1980 #48 Jul 27 2021 21:20:05
%S A001688 9,53,362,2790,24024,229080,2399760,27422640,339696000,4536362880,
%T A001688 64988179200,994447238400,16190733081600,279499828608000,
%U A001688 5100017213491200,98087346669312000,1983334021853184000,42063950934061056000,933754193111900160000
%N A001688 4th forward differences of factorial numbers A000142.
%D A001688 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001688 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001688 T. D. Noe, <a href="/A001688/b001688.txt">Table of n, a(n) for n = 0..100</a>
%H A001688 A. van Heemert, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002177749">Cyclic permutations with sequences and related problems</a>, J. Reine Angew. Math., 198 (1957), 56-72.
%H A001688 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A001688 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A001688 For n>=0 a(n) = n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9). - _Benoit Cloitre_, Jun 10 2004
%F A001688 G.f.: -log(-x+1)+1+2/(x-1)^4*x*(4-3*x+2*x^2). - _Simon Plouffe_, Master's Thesis, Uqam 1992
%F A001688 E.g.f.: (9 + 8*x + 6*x^2 + x^4)/(1 - x)^5. - _Ilya Gutkovskiy_, Jan 20 2017
%F A001688 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
%t A001688 Table[(n^4 + 6*n^3 + 17*n^2 + 20*n + 9) n!, {n, 0, 20}] (* _T. D. Noe_, Aug 09 2012 *)
%t A001688 Differences[Range[0,30]!,4] (* _Harvey P. Dale_, Jun 06 2017 *)
%o A001688 (PARI) a(n)=if(n<0,0,n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9))
%Y A001688 Cf. A000142, A001563, A001564, A001565, A001689, A095177.
%K A001688 nonn,easy
%O A001688 0,1
%A A001688 _N. J. A. Sloane_