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A092689 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, after the main diagonal is divided by 2 and the triangle is flattened, equals this flattened form of the original triangle.

%I A092689 #7 Jun 13 2017 21:56:51
%S A092689 1,1,1,3,1,3,7,5,3,7,19,13,13,7,19,51,39,33,33,19,51,141,111,99,85,89,
%T A092689 51,141,393,321,283,259,229,243,141,393,1107,925,825,747,701,627,675,
%U A092689 393,1107,3139,2675,2397,2195,2029,1929,1743,1893,1107,3139,8953,7747
%N A092689 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, after the main diagonal is divided by 2 and the triangle is flattened, equals this flattened form of the original triangle.
%C A092689 First column and main diagonal forms the central trinomial coefficients (A002426). Row sums form A092690.
%F A092689 T(n, k) = 2*T(n-1, k) + T(n-1, k+1) for 0<k<n, with T(n, n)=T(n, 0)=T(n+1, n)=A002426(n), T(0, 0)=1, T(0, 1)=T(1, 0)=1.
%e A092689 Rows begin:
%e A092689 {1},
%e A092689 {1,1},
%e A092689 {3,1,3},
%e A092689 {7,5,3,7},
%e A092689 {19,13,13,7,19},
%e A092689 {51,39,33,33,19,51},
%e A092689 {141,111,99,85,89,51,141},
%e A092689 {393,321,283,259,229,243,141,393},
%e A092689 {1107,925,825,747,701,627,675,393,1107},
%e A092689 {3139,2675,2397,2195,2029,1929,1743,1893,1107,3139},
%e A092689 {8953,7747,6989,6419,5987,5601,5379,4893,5353,3139,8953},...
%e A092689 Convolution of each row with {1,2} forms the triangle:
%e A092689 {1,2},
%e A092689 {1,3,2},
%e A092689 {3,7,5,6},
%e A092689 {7,19,13,13,14},
%e A092689 {19,51,39,33,33,38},
%e A092689 {51,141,111,99,85,89,102},
%e A092689 {141,393,321,283,259,229,243,282},...
%e A092689 which, after the main diagonal is divided by 2 and the triangle is flattened, equals the original triangle in flattened form: {1,1,1,3,1,3,7,5,3,7,19,...}.
%o A092689 (PARI) T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n-1,T(n-1,0), if(k==n,T(n,0), 2*T(n-1,k)+T(n-1,k+1))))))
%Y A092689 Cf. A002426, A092683, A092686, A092690.
%K A092689 nonn,tabl
%O A092689 0,4
%A A092689 _Paul D. Hanna_, Mar 04 2004