A001565 3rd differences of factorial numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Cf. A047920.

Cf. A180196.

Programs

GAP

List([0..20], n-> (n^3+3*n^2+5*n+2)*Factorial(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

Mathematica

Table[(n^3 +3*n^2 +5*n +2) n!, {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)
Differences[Range[0, 25]!, 3] (* Paolo Xausa, May 28 2025 *)

PARI

{a(n) = (n^3+3*n^2+5*n+2)*n!}; \\ 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

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