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A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

FORMULA

A(x;t) = Sum_{n>=0} P_n(t)*x^n = v/(1-x*t*v), where v(x;t) = A286795(x;t) and P_n(t) = Sum_{k=0..n} T(n,k)*t^k.

A000699(n+1)=T(n,0), A000330(n)=T(n,n-1), A286797(n)=P_n(1) and P_n(-1)=0 for n>0.

EXAMPLE

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]  ...

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-Fran├žois Alcover, May 23 2017, adapted from PARI *)

PROG

(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))))

CROSSREFS

Cf. A286781, A286782, A286783, A286784, A286785.

Sequence in context: A102230 A147724 A110519 * A286718 A204579 A113095

Adjacent sequences:  A286793 A286794 A286795 * A286797 A286798 A286799

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, May 21 2017

STATUS

approved

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Last modified February 22 09:40 EST 2019. Contains 320390 sequences. (Running on oeis4.)