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A121400 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for n>=k>=1, with T(0,0) = 1, T(n,n) = T(n,0) + T(n-1,n-1) for n>=1; T(n,k)=0 when n or k<0.

%I A121400 #7 Jul 22 2024 00:51:29
%S A121400 1,1,2,3,3,5,6,11,8,11,17,25,30,19,28,42,72,74,77,47,70,114,188,223,
%T A121400 198,194,117,184,302,525,609,615,509,495,301,486,827,1436,1749,1733,
%U A121400 1619,1305,1282,787,1313,2263,4012,4918,5101,4657,4206,3374,3382,2100,3576
%N A121400 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for n>=k>=1, with T(0,0) = 1, T(n,n) = T(n,0) + T(n-1,n-1) for n>=1; T(n,k)=0 when n<k or k<0.
%C A121400 Main diagonal (A121398) forms the partial sums of column 0 (A121399). The g.f. of the row sums is H(x)*(1-x)/(1-3x), where H(x) is the g.f. of column 0. This is the cascadence for function F(x) = 1 + x + x^2 when forced to form a triangle in which row n has n+1 terms for n>=0.
%F A121400 G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+y+y^2) - y ), where H(x) satisfies: H(x) = G*H(x*G)/x = g.f. of column 0 (A121399) and G/x is the g.f. of the Motzkin numbers (A001006): G = x*(1 + G + G^2).
%e A121400 Triangle begins:
%e A121400 1;
%e A121400 1, 2;
%e A121400 3, 3, 5;
%e A121400 6, 11, 8, 11;
%e A121400 17, 25, 30, 19, 28;
%e A121400 42, 72, 74, 77, 47, 70;
%e A121400 114, 188, 223, 198, 194, 117, 184;
%e A121400 302, 525, 609, 615, 509, 495, 301, 486;
%e A121400 827, 1436, 1749, 1733, 1619, 1305, 1282, 787, 1313;
%e A121400 2263, 4012, 4918, 5101, 4657, 4206, 3374, 3382, 2100, 3576;
%e A121400 6275, 11193, 14031, 14676, 13964, 12237, 10962, 8856, 9058, 5676, 9851;
%e A121400 The convolution of each row with [1,1,1] yields:
%e A121400 [1,1,1]*[1] = [1,1,1];
%e A121400 [1,1,1]*[1,2] = [1,3,3,2];
%e A121400 [1,1,1]*[3,3,5] = [3,6,11,8,5];
%e A121400 [1,1,1]*[6,11,8,11] = [6,17,25,30,19,11]; ...
%e A121400 Concatenate these convoluted rows after adding last and first terms:
%e A121400 1,1,1 + 1,3,3,2 + 3,6,11,8,5 + 6,17,25,30,19,11 + 17, ...
%e A121400 to obtain the concatenated rows of this original triangle:
%e A121400 1, 1,2, 3,3,5, 6,11,8,11, 17,25,30,19,28, ...
%o A121400 (PARI) {T(n,k)=if(n<k||k<0,0,if(n==0&k==0,1,if(n==k,T(n,0)+T(n-1,n-1), T(n-1,k-1)+T(n-1,k)+T(n-1,k+1))))}
%Y A121400 Cf. A121398 (main diagonal), A121399 (column 0), A001006 (Motzkin).
%K A121400 nonn,tabl
%O A121400 0,3
%A A121400 _Paul D. Hanna_, Jul 27 2006