

A001565


3rd differences of factorial numbers.
(Formerly M2004 N0793)


12



2, 11, 64, 426, 3216, 27240, 256320, 2656080, 30078720, 369774720, 4906137600, 69894316800, 1064341555200, 17255074636800, 296754903244800, 5396772116736000, 103484118786048000, 2086818140639232000, 44150769074700288000, 977904962186600448000
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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 j1 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).
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 5672.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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: A179120 A038725 A161947 * A199412 A074613 A247109
Adjacent sequences: A001562 A001563 A001564 * A001566 A001567 A001568


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



