OFFSET
1,4
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
a(n) is the number of strings of length n-1 defined on {0,1,2,3} that contain at least one 0, at least one 1, at least one 2, and have no restriction on the number of 3's. - Enrique Navarrete, Jan 23 2026
a(n) is the number of topologies on a set of n-1 elements with 5 open sets in the topology. - M. F. Hasler, Jun 23 2026
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).
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
a(n) = A326882(n-1, 4). - M. F. Hasler, Jun 23 2026
EXAMPLE
There are a(4) = 6 topologies with 4 nonempty open sets on a 4-1 = 3-set S = {a, b, c}: Those nonempty open sets are S and the sets which additionally have either any two singletons and their union (e.g., {a}, {b} and {a, b}), or two 2-element sets and their intersection, e.g., {a}, {a, b} and {a, c}. - M. F. Hasler, Jun 23 2026
MATHEMATICA
Table[4^(n - 1) - 3*3^(n - 1) + 3*2^(n - 1) - 1, {n, 1, 30}] (* Stefan Steinerberger, Apr 13 2006 *)
(* Alternative: *)
Table[6*StirlingS2[n, 4], {n, 1, 30}] (* G. C. Greubel, Nov 19 2017 *)
PROG
(Magma) [4^(n-1) - 3*3^(n-1) + 3*2^(n-1) - 1: n in [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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)
STATUS
approved