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 A001565 3rd differences of factorial numbers. (Formerly M2004 N0793) 15
 2, 11, 64, 426, 3216, 27240, 256320, 2656080, 30078720, 369774720, 4906137600, 69894316800, 1064341555200, 17255074636800, 296754903244800, 5396772116736000, 103484118786048000, 2086818140639232000, 44150769074700288000, 977904962186600448000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS From Emeric Deutsch, Sep 09 2010: (Start) 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). a(n) = Sum_{k>=0} k*A180196(n+2,k). (End) 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 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72. Index entries for sequences related to factorial numbers FORMULA a(n) = (n^3 + 3*n^2 + 5*n + 2)*n!. - Mitch Harris, Jul 10 2008 E.g.f.: (2 + 3*x + x^3)/(1 - x)^4. - Ilya Gutkovskiy, Jan 20 2017 MATHEMATICA Table[(n^3 +3*n^2 +5*n +2) n!, {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *) PROG (PARI) {a(n) = (n^3+3*n^2+5*n+2)*n!}; \\ G. C. Greubel, Apr 29 2019 (Magma) [(n^3+3*n^2+5*n+2)*Factorial(n): n in [0..20]]; // G. C. Greubel, Apr 29 2019 (Sage) [(n^3+3*n^2+5*n+2)*factorial(n) for n in (0..20)] # G. C. Greubel, Apr 29 2019 (GAP) List([0..20], n-> (n^3+3*n^2+5*n+2)*Factorial(n)) # G. C. Greubel, Apr 29 2019 CROSSREFS Cf. A047920. Cf. A180196. Sequence in context: A038725 A161947 A349023 * A357845 A199412 A074613 Adjacent sequences: A001562 A001563 A001564 * A001566 A001567 A001568 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified November 30 03:08 EST 2023. Contains 367452 sequences. (Running on oeis4.)