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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. 7

%I

%S 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 51,141,393,321,283,259,229,243,141,393,1107,925,825,747,701,627,675,

%U 393,1107,3139,2675,2397,2195,2029,1929,1743,1893,1107,3139,8953,7747

%N 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 First column and main diagonal forms the central trinomial coefficients (A002426). Row sums form A092690.

%F 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 Rows begin:

%e {1},

%e {1,1},

%e {3,1,3},

%e {7,5,3,7},

%e {19,13,13,7,19},

%e {51,39,33,33,19,51},

%e {141,111,99,85,89,51,141},

%e {393,321,283,259,229,243,141,393},

%e {1107,925,825,747,701,627,675,393,1107},

%e {3139,2675,2397,2195,2029,1929,1743,1893,1107,3139},

%e {8953,7747,6989,6419,5987,5601,5379,4893,5353,3139,8953},...

%e Convolution of each row with {1,2} forms the triangle:

%e {1,2},

%e {1,3,2},

%e {3,7,5,6},

%e {7,19,13,13,14},

%e {19,51,39,33,33,38},

%e {51,141,111,99,85,89,102},

%e {141,393,321,283,259,229,243,282},...

%e 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 (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 Cf. A002426, A092683, A092686, A092690.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Mar 04 2004

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