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%I M4531 N1920 #43 Jul 27 2021 04:03:14
%S 44,309,2428,21234,205056,2170680,25022880,312273360,4196666880,
%T 60451816320,929459059200,15196285843200,263309095526400,
%U 4820517384883200,92987329455820800,1885246675183872000,40080616912207872000,891690242177839104000
%N 5th forward differences of factorial numbers A000142.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001689/b001689.txt">Table of n, a(n) for n = 0..100</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H 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 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = (n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44)*n! - _Mitch Harris_, Jul 10 2008
%F E.g.f.: (44 + 45*x + 20*x^2 + 10*x^3 + x^5)/(1 - x)^6. - _Ilya Gutkovskiy_, Jan 20 2017
%F a(n) = (n+6)*a(n-1) - (n-1)*a(n-2) with a(0) = 44 and a(1) = 309. Cf. A096307. - _Peter Bala_, Jul 22 2021
%t Differences[Table[n!, {n, 0, 25}], 5] (* _T. D. Noe_, Aug 09 2012 *)
%Y Cf. A000142, A001563, A001564, A001565, A001688, A096307.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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