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
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, May 14 2017
STATUS
approved