A212857 - OEIS

%I #47 Apr 01 2024 10:30:45

%S 1,1,15,1135,271375,158408751,191740223841,429966316953825,

%T 1644839120884915215,10079117505143103766735,

%U 94135092186827772028779265,1287215725538576868883610346465,24929029117106417518788960414909025,664978827664071363541997348802227351425

%N Number of 4 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

%C From _Petros Hadjicostas_, Sep 08 2019: (Start)

%C 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=4, n, t=0).

%C 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.

%C (End)

%H Seiichi Manyama, <a href="/A212857/b212857.txt">Table of n, a(n) for n = 0..144</a> (terms n=1..19 from R. H. Hardin)

%H Morton Abramson and David Promislow, <a href="https://doi.org/10.1016/0097-3165(78)90012-2">Enumeration of arrays by column rises</a>, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) on p. 249.

%F 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 = 4. - _Petros Hadjicostas_, Sep 08 2019

%F a(n) = (n!)^4 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^4). (see Petros Hadjicostas's comment on Sep 08 2019) - _Seiichi Manyama_, Jul 18 2020

%e Some solutions for n=3:

%e   1 2 0   1 0 2   1 0 2   2 1 0   2 0 1   2 1 0   1 0 2

%e   2 1 0   1 0 2   0 2 1   0 2 1   2 1 0   1 0 2   2 1 0

%e   1 2 0   2 1 0   1 0 2   0 1 2   2 1 0   2 1 0   1 2 0

%e   2 1 0   0 1 2   2 1 0   2 1 0   1 0 2   2 1 0   2 1 0

%p A212857 := proc(n) sum(z^k/k!^4, k = 0..infinity);

%p series(%^x, z=0, n+1): n!^4*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:

%p seq(A212857(n), n=1..13); # _Peter Luschny_, May 27 2017

%t T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];

%t a[n_] := T[4, n];

%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Apr 01 2024, after _Alois P. Heinz_ in A212855 *)

%Y Row 4 of A212855.

%Y Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212858, A212859, A212860, A336196.

%K nonn

%O 0,3

%A _R. H. Hardin_, May 28 2012

%E a(0)=1 prepended by _Seiichi Manyama_, Jul 18 2020