A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 1, 4, 3, 27, 31, 5, 248, 357, 117, 7, 2830, 4742, 2218, 314, 9, 38232, 71698, 42046, 9258, 690, 11, 593859, 1216251, 837639, 243987, 30057, 1329, 13, 10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15, 202601898, 472751962, 404979234, 166620434, 35456432, 3857904, 196532, 3812, 17, 4342263000, 10651493718, 9869474106, 4561150162, 1149976242, 160594860, 11946360, 426852, 5904, 19

Offset: 0

Author

Gheorghe Coserea, May 21 2017

Keywords

• nonn
• tabf

Comments

Row n>0 contains n terms.

"The series expansion of the solution counts skeleton vertex diagrams with dressed propagators and bare interactions." (see G^2v-skeleton expansion in Molinari link)

Examples

			A(x;t) = 1 + x + (4 + 3*t)*x^2 + (27 + 31*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]      [4]      [5]    [6]   [7]
[0]  1;
[1]  1;
[2]  4,        3;
[3]  27,       31,       5;
[4]  248,      357,      117,      7;
[5]  2830,     4742,     2218,     314,     9;
[6]  38232,    71698,    42046,    9258,    690,     11;
[7]  593859,   1216251,  837639,   243987,  30057,   1329,  13;
[8]  10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15;
[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.

Crossrefs

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

Programs

Mathematica

max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = ((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 // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max - 1}] // Flatten (* 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;
};
concat(apply(p->Vecrev(p), Vec(A286795_ser(11))))
\\ test: y=A286795_ser(50); 0 == 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*y'

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 0 = 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*deriv(y,x), with y(0;t)=1, where P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.

A000699(n+1) = T(n,0), 1 = P_n(-1), A049464(n+1) = P_n(1).