A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).
1, 1, 1, 2, -1, 1, 3, 4, -3, 1, 5, -5, 10, -5, 1, 8, 15, -25, 20, -7, 1, 13, -22, 65, -65, 34, -9, 1, 21, 57, -152, 195, -133, 52, -11, 1, 34, -93, 361, -542, 461, -237, 74, -13, 1, 55, 220, -815, 1445, -1464, 935, -385, 100, -15, 1, 89, -385, 1850, -3705
Offset: 0
Examples
Triangle begins: 1; 1, 1; 2, -1, 1; 3, 4, -3, 1; 5, -5, 10, -5, 1; 8, 15, -25, 20, -7, 1; 13, -22, 65, -65, 34, -9, 1; ... Production matrix is: 1, 1; 1, -2, 1; 2, 0, -2, 1; 4, 0, 0, -2, 1; 8, 0, 0, 0, -2, 1; 16, 0, 0, 0, 0, -2, 1; 32, 0, 0, 0, 0, 0, -2, 1; 64, 0, 0, 0, 0, 0, 0, -2, 1; ...
Links
- Indranil Ghosh, Rows 0..100, flattened
Programs
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Mathematica
nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 15 2017 *)
Formula
Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A098600(n), A000032(n+1), A027961(n+1), A027974(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,k) = T(n-1,k-1) - T(n-1,k) + 3*T(n-2,k) - T(n-2,k-1) + 2*T(n-3,k) - T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = -1, T(n,k) = 0 if k<0 or if k>n.
T(n,0) = T(n-1,0) + T(n-2,0) with T(0,0) = T(1,0) = 1, T(n,k) = T(n-1,k-1) - 2*T(n-1,k) for k>=1.
G.f.: (1+2*x)/((1+2*x-y*x)*(1-x-x^2)).
Comments