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 A243920 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. 3
 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, 11, 0, 44328, 26028, 5930, 903, 108, 11, 13, 0, 755681, 435804, 96640, 14168, 1611, 154, 13, 15, 0, 14750646, 8395065, 1825600, 260484, 28566, 2607, 208, 15, 17, 0, 323999500 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Table of n, a(n) for n=0..55. FORMULA 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). 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. EXAMPLE Triangle begins: 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, 11, 0; 44328, 26028, 5930, 903, 108, 11, 13, 0; 755681, 435804, 96640, 14168, 1611, 154, 13, 15, 0; 14750646, 8395065, 1825600, 260484, 28566, 2607, 208, 15, 17, 0; 323999500, 182556012, 39122945, 5471508, 584280, 51524, 3939, 270, 17, 19, 0; ... ------------------------------------- MATRIX POWER SERIES PROPERTY. Let T equal this triangle as an infinite triangular matrix; then [I - T]^(-1) = Sum_{n>=0} T^n and equals T shifted up 1 row (with all '1's replacing the main diagonal): 1; 1, 1; 4, 3, 1; 27, 18, 5, 1; 254, 159, 40, 7, 1; 3048, 1836, 435, 70, 9, 1; 44328, 26028, 5930, 903, 108, 11, 1; ... ------------------------------------- GENERATE T FROM MATRIX POWERS OF T. Matrix square T^2 begins: 0; 0, 0; 3, 0, 0; 8, 15, 0, 0; 51, 36, 35, 0, 0; 470, 303, 80, 63, 0, 0; 5588, 3426, 835, 140, 99, 0, 0; 80904, 48060, 11150, 1743, 216, 143, 0, 0; ... so that T(3,0) = T(2,0) + [T^2](2,0) = 1 + 3 = 4; T(4,1) = T(3,1) + [T^2](3,1) = 3 + 15 = 18. Matrix cube T^3 begins: 0; 0, 0; 0, 0, 0; 15, 0, 0, 0; 71, 105, 0, 0, 0; 635, 429, 315, 0, 0, 0; 7494, 4707, 1195, 693, 0, 0, 0; 108336, 65304, 15515, 2513, 1287, 0, 0, 0; ... so that T(4,0) = T(3,0) + [T^2](3,0) + [T^3](3,0) = 4 + 8 + 15 = 27; T(5,1) = T(4,1) + [T^2](4,1) + [T^3](4,1) = 18 + 36 + 105 = 159. ------------------------------------- ILLUSTRATE G.F. FOR COLUMN k: 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)) +... 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)) +... 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)) +... ------------------------------------- PROG (PARI) {T(n, k)=if(n=c, T(r-1, c-1))))); if(n

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Last modified May 22 12:56 EDT 2024. Contains 372755 sequences. (Running on oeis4.)