A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
1, 2, 2, 1, 5, 2, 5, 9, 2, 1, 15, 14, 2, 7, 35, 20, 2, 1, 28, 70, 27, 2, 9, 84, 126, 35, 2, 1, 45, 210, 210, 44, 2, 11, 165, 462, 330, 54, 2, 1, 66, 495, 924, 495, 65, 2, 13, 286, 1287, 1716, 715, 77, 2, 1, 91, 1001, 3003, 3003, 1001, 90, 2, 15, 455, 3003, 6435, 5005, 1365, 104, 2, 1, 120, 1820, 8008, 12870, 8008, 1820, 119, 2
Offset: 1
Examples
A(x;t) = x + (2*t+2)*x^2 + (t^2+5*t+2)*x^3 + (5*t^2+9*t+2)*x^4 + ... Triangle starts: n\k [1] [2] [3] [4] [5] [6] [7] [8] [1] 1; [2] 2, 2; [3] 1, 5, 2; [4] 5, 9, 2; [5] 1, 15, 14, 2; [6] 7, 35, 20, 2; [7] 1, 28, 70, 27, 2; [8] 9, 84, 126, 35, 2; [9] 1, 45, 210, 210, 44, 2; [10] 11, 165, 462, 330, 54, 2; [11] 1, 66, 495, 924, 495, 65, 2; [12] 13, 286, 1287, 1716, 715, 77, 2; [13] 1, 91, 1001, 3003, 3003, 1001, 90, 2; [14] 15, 455, 3003, 6435, 5005, 1365, 104, 2; [15] ...
Links
- Gheorghe Coserea, Rows n = 1..202, flattened
- F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
Programs
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Mathematica
Reverse[CoefficientList[#, t]]& /@ CoefficientList[x*((1-t)*x^3 + (t^2 - 2*t - 1)*x^2 + (2*t - 1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1) + O[x]^16, x] // Rest // Flatten (* Jean-François Alcover, Feb 18 2019 *)
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PARI
N=16; x='x+O('x^N); concat(apply(p->Vec(p), Vec(Ser(x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1))))) -
PARI
N = 14; concat(1, concat(vector(N, n, Vec(substpol(((1+t)^(n+2) + (1-t)^(n+2))/2 - t^2 + 1, t^2, t)))))
Comments