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A286798 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. . 4
1, 1, 4, 2, 27, 22, 248, 264, 30, 2830, 3610, 830, 8, 38232, 55768, 18746, 1078, 593859, 961740, 414720, 46986, 576, 10401712, 18326976, 9457788, 1593664, 62682, 112, 202601898, 382706674, 226526362, 49941310, 3569882, 45296, 4342263000, 8697475368, 5740088706, 1540965514, 160998750, 4909674, 16896, 101551822350, 213865372020, 154271354280, 48205014786, 6580808784, 337737294, 4200032, 2560 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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

A000699(n+1)=T(n,0), A000108(n)=P_n(-1), A286799(n)=P_n(1).

EXAMPLE

A(x;t) = 1 + x + (4 + 2*t)*x^2 + (27 + 22*t)*x^3 + (248 + 264*t + 30*t^2)*x^4 +

Triangle starts:

n\k  [0]        [1]        [2]        [3]       [4]      [5]

[0]  1;

[1]  1;

[2]  4,         2;

[3]  27,        22;

[4]  248,       264,       30;

[5]  2830,      3610,      830,       8;

[6]  38232,     55768,     18746,     1078;

[7]  593859,    961740,    414720,    46986,    576;

[8]  10401712,  18326976,  9457788,   1593664,  62682,   112;

[9]  202601898, 382706674, 226526362, 49941310, 3569882, 45296;

[10] ...

MATHEMATICA

max = 12; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = 1 + x y0[x, t]^2 + 3 t x^3 y0[x, t]^2 D[y0[x, t], x] + x^2 (2 y0[x, t] D[y0[x, t], x] + t (2 y0[x, t]^3 - D[y0[x, t], x] + y0[x, t] D[y0[x, t], x])) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];

P[n_, t_] := Coefficient[y0[x, t] , x, n];

row[n_] := CoefficientList[P[n, t], t];

Table[row[n], {n, 0, max}] // Flatten (* Jean-Fran├žois 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)));

};

concat(apply(p->Vecrev(p), Vec(A286798_ser(12))))

\\ test: y=A286798_ser(50); x^2*y' == (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2)

CROSSREFS

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

Sequence in context: A058167 A140331 A095896 * A123670 A200032 A121667

Adjacent sequences:  A286795 A286796 A286797 * A286799 A286800 A286801

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, May 21 2017

STATUS

approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)