A346385 - OEIS

%I #16 Aug 17 2021 02:24:18

%S 1,3,19,151,1439,16651,234651,3966271,78504063,1778555587,45302809003,

%T 1279960719335,39697452556959,1340332692660027,48929424425580219,

%U 1920103548827941263,80597817202971009535,3603262730476776975731,170923354522784683176267

%N a(n) = f(n,n) where f(0,n) = f(n,0) = n^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^n instead of 1.

%C a(n+1)/a(n) ~ e*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).

%t F[0, 0] = 1; F[m_, 0] := m!; F[0, n_] := 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. A000312, A001850, A008288, A344576, A346374.

%K nonn

%O 0,2

%A _José María Grau Ribas_, Jul 14 2021