A001565 3rd differences of factorial numbers. (Formerly M2004 N0793)

2, 11, 64, 426, 3216, 27240, 256320, 2656080, 30078720, 369774720, 4906137600, 69894316800, 1064341555200, 17255074636800, 296754903244800, 5396772116736000, 103484118786048000, 2086818140639232000, 44150769074700288000, 977904962186600448000

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