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Irregular triangular array: T(n,0) = 1 for n >= 1; T(n,1) = T(n,2) = 1 for n >= 1; T(n,k) = T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) for 3 <= k <= 2n-1 and n >= 2; and T(n,2n) = T(n-1,2n-3) + T(n-1,2n-2) for n >= 2.
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%I #20 Jun 24 2020 14:15:08

%S 1,1,1,1,1,1,1,3,2,1,1,1,3,5,6,5,1,1,1,3,5,9,14,16,11,1,1,1,3,5,9,17,

%T 28,39,41,27,1,1,1,3,5,9,17,31,54,84,108,107,68,1,1,1,3,5,9,17,31,57,

%U 102,169,246,299,283,175,1,1,1,3,5,9,17,31,57

%N Irregular triangular array: T(n,0) = 1 for n >= 1; T(n,1) = T(n,2) = 1 for n >= 1; T(n,k) = T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) for 3 <= k <= 2n-1 and n >= 2; and T(n,2n) = T(n-1,2n-3) + T(n-1,2n-2) for n >= 2.

%H R. J. Mathar, <a href="/A027082/b027082.txt">Table of n, a(n) for n = 0..1000</a>, recomputing the old b-file for offset 0.

%e The first 6 rows:

%e 1

%e 1 .. 1 .. 1

%e 1 .. 1 .. 1 .. 3 .. 2

%e 1 .. 1 .. 1 .. 3 .. 5 .. 6 .. 5

%e 1 .. 1 .. 1 .. 3 .. 5 .. 9 .. 14 .. 16 .. 11

%e 1 .. 1 .. 1 .. 3 .. 5 .. 9 .. 17 .. 28 .. 39 .. 41 .. 27

%p A027082 := proc(n,k)

%p option remember;

%p if k < 0 or k > 2*n then

%p 0;

%p elif k <=2 then

%p 1;

%p elif k = 2*n then

%p procname(n-1,k-3)+procname(n-1,k-2) ;

%p else

%p procname(n-1,k-3)+procname(n-1,k-2)+procname(n-1,k-1) ;

%p end if;

%p end proc:

%p seq(seq(A027082(n,k),k=0..2*n),n=0..10) ; # _R. J. Mathar_, Jun 24 2020

%t t[n_, 0] := 1; t[n_, 1] := 1; t[n_, 2] := 1; t[n_, k_] := t[n, k] = If[k < 2 n, t[n - 1, k - 3] + t[n - 1, k - 2] + t[n - 1, k - 1], t[n - 1, 2 n - 3] + t[n - 1, 2 n - 2]]; u = Table[t[n, k], {n, 0, 10}, {k, 0, 2 n}]; v = Flatten[u] (* A027082 *)

%K nonn,tabf

%O 0,8

%A _Clark Kimberling_

%E Corrected and edited by _Clark Kimberling_, Aug 06 2014

%E Offset corrected to 0 and b-file recomputed by _R. J. Mathar_, Jun 24 2020