OFFSET
0,3
COMMENTS
From Petros Hadjicostas, Sep 08 2019: (Start)
We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=7, n, t=0).
Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.
(End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..92 (terms n=1..19 from R. H. Hardin)
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) on p. 249.
FORMULA
a(n) = (-1)^(n-1) + Sum_{s = 1..n-1} a(s) * (-1)^(n-s-1) * binomial(n,s)^m for n >= 2 with a(1) = 1. Here m = 7. - Petros Hadjicostas, Sep 08 2019
a(n) = (n!)^7 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^7). (see Petros Hadjicostas's comment on Sep 08 2019) - Seiichi Manyama, Jul 18 2020
EXAMPLE
Some solutions for n=3:
0 1 2 0 1 2 0 2 1 0 1 2 0 2 1 0 2 1 0 2 1
1 2 0 0 2 1 0 2 1 1 0 2 0 2 1 1 0 2 2 1 0
1 0 2 2 1 0 2 0 1 0 1 2 2 0 1 1 0 2 1 2 0
0 2 1 1 0 2 0 2 1 1 0 2 0 1 2 2 0 1 0 1 2
2 0 1 2 1 0 1 0 2 2 1 0 1 2 0 0 1 2 1 2 0
2 1 0 0 1 2 1 0 2 0 1 2 2 0 1 1 0 2 2 1 0
1 2 0 2 1 0 0 1 2 0 2 1 2 1 0 2 0 1 2 0 1
MAPLE
A212860 := proc(n) sum(z^k/k!^7, k = 0..infinity);
series(%^x, z=0, n+1): n!^7*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A212860(n), n=1..10); # Peter Luschny, May 27 2017
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[7, n];
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 28 2012
EXTENSIONS
a(0)=1 prepended by Seiichi Manyama, Jul 18 2020
STATUS
approved