A346374 - OEIS

%I #38 Jul 20 2021 03:10:55

%S 1,3,15,85,511,3221,21339,149969,1133215,9343525,85089883,860979329,

%T 9665236335,119534610885,1613648163963,23564301925713,369437215969599,

%U 6180713141601765,109812899448776475,2063836219057694753,40894619124091715983,851891564231714448133

%N a(n) = f(n,n) where f(0,n) = f(n,0) = n! and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

%C Main diagonal of square array of Delannoy-like numbers, but with the borders given by n! instead of 1.

%C All terms are odd.

%C a(n+1)/a(n) ~ n (conjectured).

%F Conjecture:

%F 0 = (310000 - 87575*n^2 + 25110*n^3 - 2015*n^4)*a(n-6)

%F + (-3082368 + 190872*n + 1063326*n^2 - 378291*n^3 + 37173*n^4)*a(n-5)

%F + (8418604 - 1615685*n - 3834272*n^2 + 1836495*n^3 - 232486*n^4)*a(n-4)

%F + (-4087868 + 3360702*n + 3358400*n^2 - 2858550*n^3 + 517914*n^4)*a(n-3)

%F + (-3408770 - 992468*n + 1472981*n^2 + 176556*n^3 - 157931*n^4)*a(n-2)

%F + (582108 + 205634*n - 541582*n^2 + 120633*n^3 + 12993*n^4)*a(n-1)

%F + (-3362 - 20943*n + 37298*n^2 - 12993*n^3)*a(n).

%p b:= proc(x, y) option remember; `if`(x=0, y!,

%p       b(x-1, y)+b(sort([x, y-1])[])+b(x-1, y-1))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..26);  # _Alois P. Heinz_, Jul 14 2021

%t F[0, 0] = 1; F[m_, 0] := m!; F[0, n_] := n!;

%t F[m_, n_] := F[m, n] =   F[m - 1 , n ] + F[m , n - 1] +  F[m - 1, n - 1];

%t Table[F[n, n], {n, 0, 100}]

%Y Cf. A000142, A001850, A008288, A344576.

%K nonn

%O 0,2

%A _José María Grau Ribas_, Jul 14 2021