A286781 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 2, 1, 10, 9, 1, 74, 91, 23, 1, 706, 1063, 416, 46, 1, 8162, 14193, 7344, 1350, 80, 1, 110410, 213953, 134613, 34362, 3550, 127, 1, 1708394, 3602891, 2620379, 842751, 125195, 8085, 189, 1, 29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1, 576037442, 1375636129, 1223392968, 533394516, 124266346, 15653598, 1023340, 31356, 366, 1

Offset

0,2

Comments

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 self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).

Links

Gheorghe Coserea, Rows n=0..122, flattened

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

Formula

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

A000698(n+1)=T(n,0), A101986(n)=T(n,n-1), A000108(n)=P_n(-1), A286794(n)=P_n(1).

Example

A(x;t) = 1 + (2 + t)*x + (10 + 9*t + t^2)*x^2 + (74 + 91*t + 23*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]  [8]
[0]  1;
[1]  2,        1;
[2]  10,       9,        1;
[3]  74,       91,       23,       1;
[4]  706,      1063,     416,      46,       1;
[5]  8162,     14193,    7344,     1350,     80,      1;
[6]  110410,   213953,   134613,   34362,    3550,    127,    1;
[7]  1708394,  3602891,  2620379,  842751,   125195,  8085,   189,   1;
[8]  29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1;
[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_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max-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;
};
concat(apply(p->Vecrev(p), Vec(A286781_ser(10))))
\\ test: y = A286781_ser(50); y*(1-x*y)^2 == (1 + x*y + 2*x^2*deriv(y, 'x)) * (1 - x*y*(1-t))

Crossrefs

For vertex and polarization functions see A286782 and A286783. For GWA of the self-energy and polarization functions see A286784 and A286785.

Columns k=0-8 give: A000698(k=0), A286786(k=1), A286787(k=2), A286788(k=3), A286789(k=4), A286790(k=5), A286791(k=6), A286792(k=7), A286793(k=8).

Sequence in context: A217108 A127259 A152260 * A193727 A138098 A081098

Adjacent sequences: A286778 A286779 A286780 * A286782 A286783 A286784

Keyword

nonn,tabl

Author

Gheorghe Coserea, May 14 2017

Status

approved