A110112 - OEIS

%I #15 Dec 27 2019 11:30:45

%S 1,1,1,1,3,1,1,15,5,1,1,60,55,7,1,1,260,385,133,9,1,1,1092,3311,1330,

%T 261,11,1,1,4641,25585,18430,3393,451,13,1,1,19635,208335,210490,

%U 68237,7216,715,15,1,1,83215,1652145,2673223,1037673,197456,13585,1065,17,1,1

%N Square array of numbers associated to the recurrences b(k) = b(k-1) + n*b(k-2); array T(n,k), read by descending antidiagonals, for n, k >= 0.

%C Rows include A001655, (-1)^n*A015266(n+3), A110111.

%F T(n, k) = a(n, k+1) * a(n, k+2) * a(n, k+3)/(n+1), where a(n, k) is the solution to a(n, k) = a(n, k-1) + n*a(n, k-2) for k >= 2 with a(n, 0) = 0 and a(n, 1) = 1 for all n >= 0.

%F Row n has g.f. 1/((1 + n*x - n^3*x^2) * (1 - (3*n + 1)*x - n^3*x^2)).

%e Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:

%e   1,  1,   1,    1,      1,       1,        1,          1, ...

%e   1,  3,  15,   60,    260,    1092,     4641,      19635, ...

%e   1,  5,  55,  385,   3311,   25585,   208335,    1652145, ...

%e   1,  7, 133, 1330,  18430,  210490,  2673223,   31940881, ...

%e   1,  9, 261, 3393,  68237, 1037673, 18598293,  300963537, ...

%e   1, 11, 451, 7216, 197456, 3761296, 89565861, 1842200151, ...

%e   ...

%p a := proc(n, k) local v; option remember; if k = 0 and 0 <= n then v := 0; end if; if k = 1 and 0 <= n then v := 1; end if; if 2 <= k and 0 <= n then v := a(n, k - 1) + n*a(n, k - 2); end if; v; end proc;

%p T := proc(n, k) a(n, k + 1)*a(n, k + 2)*a(n, k + 3)/(n + 1); end proc;

%p seq(seq(T(k,n-k), k=0..n), n=0..10); # _Petros Hadjicostas_, Dec 26 2019

%Y Cf. A083856.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Jul 12 2005