A028244 a(n) = 4^(n-1) - 3*3^(n-1) + 3*2^(n-1) - 1 (essentially Stirling numbers of second kind).

0, 0, 0, 6, 60, 390, 2100, 10206, 46620, 204630, 874500, 3669006, 15195180, 62350470, 254135700, 1030793406, 4166023740, 16792841910, 67558001700, 271392695406, 1089054420300, 4366671742950, 17498055448500, 70086339807006

Offset: 1

Author

N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)

Keywords

• nonn
• easy

Comments

For n>=4, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1661
• K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters ( Volume: 50, Issue: 1, January 2 2014 ).
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (10, -35, 50, -24).

Crossrefs

Cf. A000453, A008277, A032263, A163626.

Programs

Magma

[4^(n-1) - 3*3^(n-1) + 3*2^(n-1) - 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

Mathematica

Table[4^(n - 1) - 3*3^(n - 1) + 3*2^(n - 1) - 1, {n, 1, 30}] (* Stefan Steinerberger, Apr 13 2006 *)
Table[6*StirlingS2[n,4], {n,1,30}] (* G. C. Greubel, Nov 19 2017 *)

PARI

for(n=1,30, print1(6*stirling(n,4,2), ", ")) \\ G. C. Greubel, Nov 19 2017

Formula

a(n) = 6*S(n, 4) = 6*A000453(n). - Emeric Deutsch, May 02 2004

G.f.: 6x^4/((1-x)(1-2x)(1-3x)(1-4x)). - R. J. Mathar, Oct 23 2008

E.g.f.: (exp(4*x) - 4*exp(3*x) + 6*exp(2*x) - 4*exp(x) + 1)/4, with a(0) = 0. - Wolfdieter Lang, May 03 2017

a(n) = 2*A032263(n). - Alois P. Heinz, Jan 24 2018