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%I M2004 N0793 #37 Sep 08 2022 08:44:29
%S 2,11,64,426,3216,27240,256320,2656080,30078720,369774720,4906137600,
%T 69894316800,1064341555200,17255074636800,296754903244800,
%U 5396772116736000,103484118786048000,2086818140639232000,44150769074700288000,977904962186600448000
%N 3rd differences of factorial numbers.
%C From _Emeric Deutsch_, Sep 09 2010: (Start)
%C a(n) is the number of isolated entries in all permutations of [n+2]. An entry j of a permutation p is isolated if it is not preceded by j-1 and not followed by j+1. For example, the permutation 23178564 has 2 isolated entries: 1 and 4. a(1)=11 because in 123, 1'3'2', 2'1'3', 231', 3'12, and 3'2'1' we have a total of 11 isolated entries (they are marked).
%C a(n) = Sum_{k>=0} k*A180196(n+2,k). (End)
%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="/A001565/b001565.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^3 + 3*n^2 + 5*n + 2)*n!. - _Mitch Harris_, Jul 10 2008
%F E.g.f.: (2 + 3*x + x^3)/(1 - x)^4. - _Ilya Gutkovskiy_, Jan 20 2017
%t Table[(n^3 +3*n^2 +5*n +2) n!, {n, 0, 20}] (* _T. D. Noe_, Aug 09 2012 *)
%o (PARI) {a(n) = (n^3+3*n^2+5*n+2)*n!}; \\ _G. C. Greubel_, Apr 29 2019
%o (Magma) [(n^3+3*n^2+5*n+2)*Factorial(n): n in [0..20]]; // _G. C. Greubel_, Apr 29 2019
%o (Sage) [(n^3+3*n^2+5*n+2)*factorial(n) for n in (0..20)] # _G. C. Greubel_, Apr 29 2019
%o (GAP) List([0..20], n-> (n^3+3*n^2+5*n+2)*Factorial(n)) # _G. C. Greubel_, Apr 29 2019
%Y Cf. A047920.
%Y Cf. A180196.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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