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%I #11 Apr 26 2016 12:44:36
%S 0,1,0,1,3,0,4,3,5,0,27,18,5,7,0,254,159,40,7,9,0,3048,1836,435,70,9,
%T 11,0,44328,26028,5930,903,108,11,13,0,755681,435804,96640,14168,1611,
%U 154,13,15,0,14750646,8395065,1825600,260484,28566,2607,208,15,17,0,323999500
%N Triangle T, read by rows, where T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = 2*n+1 and T(n,n)=0 for n>=0, where T^j denotes the j-th matrix power of T.
%F G.f. of column k: (2*k+1)*x^(k+1) = Sum_{n>=0} T(n,k) * x^n * (1-x)^(n-k) / Product_{j=k+1..n-1} (1+2*j*x).
%F T(n,k) = [x^n] { (2*k+1)*x^(k+1) - Sum_{m=k+1..n-1} T(m,k)*x^m*(1-x)^(m-k) / Product_{j=k+1..m-1} (1+2*j*x) } for n>k with T(n,k)=0 when k>=n.
%e Triangle begins:
%e 0;
%e 1, 0;
%e 1, 3, 0;
%e 4, 3, 5, 0;
%e 27, 18, 5, 7, 0;
%e 254, 159, 40, 7, 9, 0;
%e 3048, 1836, 435, 70, 9, 11, 0;
%e 44328, 26028, 5930, 903, 108, 11, 13, 0;
%e 755681, 435804, 96640, 14168, 1611, 154, 13, 15, 0;
%e 14750646, 8395065, 1825600, 260484, 28566, 2607, 208, 15, 17, 0;
%e 323999500, 182556012, 39122945, 5471508, 584280, 51524, 3939, 270, 17, 19, 0; ...
%e -------------------------------------
%e MATRIX POWER SERIES PROPERTY.
%e Let T equal this triangle as an infinite triangular matrix; then
%e [I - T]^(-1) = Sum_{n>=0} T^n and equals T shifted up 1 row
%e (with all '1's replacing the main diagonal):
%e 1;
%e 1, 1;
%e 4, 3, 1;
%e 27, 18, 5, 1;
%e 254, 159, 40, 7, 1;
%e 3048, 1836, 435, 70, 9, 1;
%e 44328, 26028, 5930, 903, 108, 11, 1; ...
%e -------------------------------------
%e GENERATE T FROM MATRIX POWERS OF T.
%e Matrix square T^2 begins:
%e 0;
%e 0, 0;
%e 3, 0, 0;
%e 8, 15, 0, 0;
%e 51, 36, 35, 0, 0;
%e 470, 303, 80, 63, 0, 0;
%e 5588, 3426, 835, 140, 99, 0, 0;
%e 80904, 48060, 11150, 1743, 216, 143, 0, 0; ...
%e so that
%e T(3,0) = T(2,0) + [T^2](2,0) = 1 + 3 = 4;
%e T(4,1) = T(3,1) + [T^2](3,1) = 3 + 15 = 18.
%e Matrix cube T^3 begins:
%e 0;
%e 0, 0;
%e 0, 0, 0;
%e 15, 0, 0, 0;
%e 71, 105, 0, 0, 0;
%e 635, 429, 315, 0, 0, 0;
%e 7494, 4707, 1195, 693, 0, 0, 0;
%e 108336, 65304, 15515, 2513, 1287, 0, 0, 0; ...
%e so that
%e T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 4 + 8 + 15 = 27;
%e T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 18 + 36 + 105 = 159.
%e -------------------------------------
%e ILLUSTRATE G.F. FOR COLUMN k:
%e k=0: 1*x = T(1,0)*x*(1-x) + T(2,0)*x^2*(1-x)^2/(1+2*1*x) + T(3,0)*x^3*(1-x)^3/((1+2*1*x)*(1+2*2*x)) + T(4,0)*x^4*(1-x)^4/((1+2*1*x)*(1+2*2*x)*(1+2*3*x)) +...
%e k=1: 3*x^2 = T(2,1)*x^2*(1-x) + T(3,1)*x^3*(1-x)^2/(1+2*2*x) + T(4,1)*x^4*(1-x)^3/((1+2*2*x)*(1+2*3*x)) + T(5,1)*x^5*(1-x)^4/((1+2*2*x)*(1+2*3*x)*(1+2*4*x)) +...
%e k=2: 5*x^3 = T(3,2)*x^3*(1-x) + T(4,2)*x^4*(1-x)^2/(1+2*3*x) + T(5,2)*x^5*(1-x)^3/((1+2*3*x)*(1+2*4*x)) + T(6,2)*x^6*(1-x)^4/((1+2*3*x)*(1+2*4*x)*(1+2*5*x)) +...
%e -------------------------------------
%o (PARI) {T(n, k)=if(n<k+1, 0, polcoeff((2*k+1)*x^(k+1)-sum(m=k+1, n-1, T(m, k)*x^m*(1-x)^(m-k)/prod(j=k+1, m-1, 1+2*j*x+x*O(x^n))), n))}
%o for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o (PARI) /* Using the matrix power formula: */
%o {T(n, k)=local(M=if(n<=0, Mat(1), matrix(n, n, r, c, if(r>=c, T(r-1, c-1))))); if(n<k|k<0, 0, if(n==k, 0, if(n==k+1, 2*n-1, sum(j=1, n-k-1, (M^j)[n, k+1]) )))}
%o for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y Cf. A132623 (variant), A243696 (column 0), A243921, A243922, A243923.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jun 15 2014
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