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%I #4 Jun 14 2017 00:31:05
%S 1,1,1,1,1,2,2,2,3,4,4,4,6,8,11,11,11,15,19,25,33,33,33,44,55,70,89,
%T 114,114,114,147,180,224,279,349,438,438,438,552,666,813,993,1217,
%U 1496,1845,1845,1845,2283,2721,3273,3939,4752,5745,6962,8458,8458,8458,10303
%N Triangle, read by rows, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>=k>=2, with T(n+1,1) = T(n+1,0) = T(n,n) and T(0,0) = 1 for n>=0.
%C G.f. of column 0 (A127782) satisfies: G(x) = 1 + x*G(x+x^2).
%F T(n,0) = Sum_{k=0..[n/2]} C(n-k,k)*T(n-k-1,0) for n>0 with T(0,0)=1. For column 1, T(n,1) = Sum_{k=0..[n/2]+1} [C(n-k,k) + C(n-k+1,k-1)]*T(n-k-1,1) for n>=2, with T(0,1)=T(1,1)=1.
%e T(5,3) = T(5,2) + T(4,1) = 15 + 4 = 19;
%e T(6,4) = T(6,3) + T(5,2) = 55 + 15 = 70;
%e T(7,0) = T(6,6) = 89 + 25 = 114.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 2, 2, 3, 4;
%e 4, 4, 6, 8, 11;
%e 11, 11, 15, 19, 25, 33;
%e 33, 33, 44, 55, 70, 89, 114;
%e 114, 114, 147, 180, 224, 279, 349, 438;
%e 438, 438, 552, 666, 813, 993, 1217, 1496, 1845;
%e 1845, 1845, 2283, 2721, 3273, 3939, 4752, 5745, 6962, 8458; ...
%o (PARI) T(n,k)=if(n<k || k<0,0,if(k==0,if(n==0,1,T(n-1,n-1)),T(n,k-1)+T(n-1,k-2)))
%Y Cf. A127782 (column 0), A130522 (diagonal).
%K nonn,tabl
%O 0,6
%A _Paul D. Hanna_, Jun 02 2007
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