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A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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)

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 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).

EXAMPLE

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] ...

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-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;

};

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'

CROSSREFS

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

Sequence in context: A189742 A243237 A072044 * A127138 A064081 A211364

Adjacent sequences:  A286792 A286793 A286794 * A286796 A286797 A286798

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.)