A287040 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
0, 1, 1, 1, 1, 2, 5, 3, 2, 8, 17, 22, 10, 5, 29, 91, 106, 94, 35, 14, 140, 431, 701, 582, 396, 126, 42, 661, 2501, 4067, 4544, 2980, 1654, 462, 132, 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429, 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430, 120909, 550811, 1245517, 1727148, 1776310, 1291654, 772422, 322204, 117016, 24310, 4862
Offset: 0
Examples
A(x;t) = t + (1 + t + t^2)*x + (2 + 5*t + 3*t^2 + 2*t^3)*x^2 + (8 + 17*t + 22*t^2 + 10*t^3 + 5*t^4)*x^3 + ... Triangle starts: n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [0] 0, 1; [1] 1, 1, 1; [2] 2, 5, 3, 2; [3] 8, 17, 22, 10, 5; [4] 29, 91, 106, 94, 35, 14; [5] 140, 431, 701, 582, 396, 126, 42; [6] 661, 2501, 4067, 4544, 2980, 1654, 462, 132; [7] 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429; [8] 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430; [9] ...
Links
- Gheorghe Coserea, Rows n=0..200, flattened
- Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017.
Programs
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Mathematica
nmax = 10; y[0, t_] := t; y[, ] = 0; Do[y[x_, t_] = Series[t + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, nmax}, {t, 0, nmax}] // Normal, {n, 0, nmax}]; CoefficientList[#, t]& /@ CoefficientList[y[x, t]+O[x]^nmax, x] // Flatten (* Jean-François Alcover, Dec 13 2018 *)
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PARI
A287040_ser(N) = { my(x='x+O('x^N), t='t, F0=t, F1=0, n=1); while(n++, F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0; if (F1 == F0, break()); F0 = F1; ); F0; }; concat(apply(p->Vecrev(p), Vec(A287040_ser(10)))) \\ test: y=A287040_ser(50); y == t + x*y^2 + x*deriv(y, t) + x*y