OFFSET
1,2
COMMENTS
Column 4 of A212855.
From Petros Hadjicostas, Aug 25 2019:
All formulas below follow from the theory in the documentation of array A309951.
We have Sum_{s = 0..A000041(4)} (-1)^s * A309951(4,s) * a(n-s) = 0, i.e., a(n) - 47*a(n-1) + 718*a(n-2) - 4416*a(n-3) + 10656*a(n-4) - 6912*a(n-5) = 0 for n >= 6. This is a consequence of Eq. (6) on p. 248 of Abramson and Promislow (1978).
Note that in R. J. Mathar's formula a(n) = 24^n + 6^n - 3*12^n + 2*4^n - 1^n, the numbers 1, 4, 12, 6, and 24 (that are raised to the n-th power) are the multinomial coefficients of the A000041(4) = 5 integer partitions of 4: 4!/4! = 1, 4!/(1!3!) = 4, 12 = 4!/(1!1!2!), 6 = 4!/(2!2!), 24 = 4!/(1!1!1!1!).
Note also that these numbers appear also in the denominator of the Colin Barker's g.f.: (1 - x)*(1 - 4*x)*(1 - 6*x)*(1 - 12*x)*(1 - 24*x) = 1 - 47*x + 718*x^2 - 4416*x^3 + 10656*x^4 - 6912*x^5.
(End)
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..210
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6) on p. 248 (set t:=0).
FORMULA
Empirical: a(n) = 47*a(n-1) - 718*a(n-2) + 4416*a(n-3) - 10656*a(n-4) + 6912*a(n-5).
Empirical: a(n) = 24^n + 6^n - 3*12^n + 2*4^n - 1. R. J. Mathar, Jun 25 2012
Empirical g.f.: x*(1 + 164*x - 216*x^2 - 3744*x^3) / ((1 - x)*(1 - 4*x)*(1 - 6*x)*(1 - 12*x)*(1 - 24*x)). - Colin Barker, Jul 21 2018
EXAMPLE
Some solutions for n=3:
..1..3..0..2....3..1..2..0....1..2..0..3....1..2..0..3....1..2..0..3
..2..1..0..3....3..1..0..2....0..1..3..2....3..0..2..1....2..1..3..0
..2..3..1..0....1..2..0..3....3..2..0..1....1..2..0..3....1..3..2..0
MATHEMATICA
T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k-j], {j, 1, k}]];
a[n_] := T[n, 4];
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 28 2012
STATUS
approved