A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
1, 1, 1, 4, 5, 1, 27, 40, 14, 1, 248, 419, 200, 30, 1, 2830, 5308, 3124, 700, 55, 1, 38232, 78070, 53620, 15652, 1960, 91, 1, 593859, 1301088, 1007292, 356048, 60550, 4704, 140, 1, 10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1, 202601898, 495263284, 456715752, 209878440, 52619854, 7322172, 545908, 19800, 285, 1, 4342263000, 11085720018, 10921213644, 5516785032, 1579263840, 264576774, 25677652, 1372228, 36300, 385, 1
Offset: 0
Examples
A(x;t) = 1 + (1 + t)*x + (4 + 5*t + t^2)*x^2 + (27 + 40*t + 14*t^2 + t^3)*x^3 + ... Triangle starts: n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [0] 1; [1] 1; 1; [2] 4, 5, 1; [3] 27, 40, 14, 1; [4] 248, 419, 200, 30, 1; [5] 2830, 5308, 3124, 700, 55, 1; [6] 38232, 78070, 53620, 15652, 1960, 91, 1; [7] 593859, 1301088, 1007292, 356048, 60550, 4704, 140, 1; [8] 10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1; [9] ...
Links
- Gheorghe Coserea, Rows n=0..123, flattened
- Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
Programs
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Mathematica
max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1 + x*(1 + 2*t + x*t^2)*y0[x, t]^2 + t*(1 - t)*x^2*y0[x, t]^3 + 2*x^2*y0[x, t]*D[y0[x, t], x])/(1 + 2*x*t) + O[x]^n]; y0[x_, t_] = y1[x, t]]; row[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t]; Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 23 2017, adapted from PARI *) -
PARI
A286795_ser(N, t='t) = { my(x='x+O('x^N), y0=1, y1=0, n=1); while(n++, y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0'); y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0; }; A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v); concat(apply(p->Vecrev(p), Vec(A286796_ser(11))))