A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 1, 6, 3, 50, 45, 5, 518, 637, 161, 7, 6354, 9567, 3744, 414, 9, 89782, 156123, 80784, 14850, 880, 11, 1435330, 2781389, 1749969, 446706, 46150, 1651, 13, 25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835, 15, 505785122, 1141864959, 928825464, 354665628, 68167144, 6500086, 281792, 4556, 17, 10944711398, 26137086451, 23244466392, 10134495804, 2361060574, 297418362, 19443460, 595764, 6954, 19

Offset

0,3

Comments

Row n>0 contains n terms.

T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the vertex function in a many-body theory of fermions with two-body interaction (see Molinari link).

Links

Gheorghe Coserea, Rows n=0..123, flattened

Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

Formula

A(x;t) = Sum_{n>=0} P_n(t)*x^n = 1 + x*s + 2*x^2 * deriv(s,x), where s(x;t) = A286781(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.

T(n+1,k) = (2*n+1)*A286781(n,k), A005416(n)=T(n,0), A088218(n)=P_n(-1).

Example

A(x;t) = 1 + x + (6 + 3*t)*x^2 + (50 + 45*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]
[0]  1;
[1]  1;
[2]  6,        3;
[3]  50,       45,       5;
[4]  518,      637,      161,      7;
[5]  6354,     9567,     3744,     414,      9;
[6]  89782,    156123,   80784,    14850,    880,     11;
[7]  1435330,  2781389,  1749969,  446706,   46150,   1651,   13;
[8]  25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835,  15;
[9]  ...

Mathematica

max = 10; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = (1 + x*y0[x, t] + 2*x^2*D[y0[x, t], x])*(1 - x*y0[x, t]*(1 - t))/(1 - x*y0[x, t])^2 + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := (2n+1) CoefficientList[Coefficient[y0[x, t], x, n], t];
T[0, 0] = 1; T[n_, k_] := row[n-1][[k+1]];
Table[T[n, k], {n, 0, max}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)

Prog

(PARI)
A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1; );
  y0;
};
A286782_ser(N, t='t) = my(s=A286781_ser(N, t)); 1 + x*s + 2*x^2 * deriv(s, 'x);
concat(apply(p->Vecrev(p), Vec(A286782_ser(10))))

Crossrefs

Sequence in context: A229130 A349492 A088390 * A054380 A348171 A348142

Adjacent sequences: A286779 A286780 A286781 * A286783 A286784 A286785

Keyword

nonn,tabf

Author

Gheorghe Coserea, May 14 2017

Status

approved