A286800 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 1, 2, 7, 6, 63, 74, 10, 729, 974, 254, 8, 10113, 15084, 5376, 406, 161935, 264724, 117424, 14954, 320, 2923135, 5163276, 2697804, 481222, 23670, 112, 58547761, 110483028, 65662932, 14892090, 1186362, 21936, 1286468225, 2570021310, 1695874928, 461501018, 51034896, 1866986, 11264, 30747331223, 64547199082, 46461697760, 14603254902, 2055851560, 116329886, 1905888, 2560

Offset

1,3

Comments

Row n>0 contains floor(2*(n+1)/3) terms.

Links

Gheorghe Coserea, Rows n=1..123, flattened

Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Formula

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

A049464(n) = T(n,0), P_n(-1) = (-1)^(n-1), A287029(n) = P_n(1).

Example

A(x;t) = x + (1 + 2*t)*x^2 + (7 + 6*t)*x^3 + (63 + 74*t + 10*t^2)*x^4 + ...
Triangle starts:
n\k  [0]       [1]        [2]       [3]       [4]      [5]
[1]  1;
[2]  1,        2;
[3]  7,        6;
[4]  63,       74,        10;
[5]  729,      974,       254,      8;
[6]  10113,    15084,     5376,     406;
[7]  161935,   264724,    117424,   14954,    320;
[8]  2923135,  5163276,   2697804,  481222,   23670,   112;
[9]  58547761, 110483028, 65662932, 14892090, 1186362, 21936;
[10] ...

Mathematica

max = 12; y0[0, _] = y1[0, _] = 0; y0[x_, t_] = x; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1/(-1 + y0[x, t]))*x*(-1 - y0[x, t]^2 - 2*y0[x, t]*(-1 + D[y0[x, t], x]) + t*x*(-1 + y0[x, t])*(2*(-1 + y0[x, t])^2 + (x*(-1 + y0[x, t]) + y0[x, t])*D[y0[x, t], x])) + O[x]^n]; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[SeriesCoefficient[y0[x, t], {x, 0, n}], t];
Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 24 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;
};
A286798_ser(N, t='t) = {
  my(v = A286795_ser(N, t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
A286800_ser(N, t='t) = {
  my(v = A286798_ser(N, t)); 1-1/subst(v, 'x, serreverse(x*v^2));
};
concat(apply(p->Vecrev(p), Vec(A286800_ser(12))))
\\ test: y=A286800_ser(50); x*y' == (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y)

Crossrefs

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

Sequence in context: A211368 A225101 A351823 * A323722 A021365 A179378

Adjacent sequences: A286797 A286798 A286799 * A286801 A286802 A286803

Keyword

nonn,tabf

Author

Gheorghe Coserea, May 22 2017

Status

approved