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%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
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