A286783 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 3, 15, 5, 105, 77, 7, 945, 1044, 234, 9, 10395, 14784, 5390, 550, 11, 135135, 227877, 113126, 19760, 1105, 13, 2027025, 3862305, 2371845, 586425, 58275, 1995, 15, 34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17, 654729075, 1469813400, 1185214452, 446964322, 84487110, 7888876, 333564, 5244, 19, 13749310575, 32718512925, 28937407212, 12516198870, 2884205268, 358182846, 23006928, 690480, 7875, 21

Offset: 0

Author

Gheorghe Coserea, May 14 2017

Keywords

• nonn
• tabf

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

Examples

			A(x;t) = 1 + 3*x + (15 + 5*t)*x^2 + (105 + 77*t + 7*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]   [7]
[0]  1;
[1]  3;
[2]  15,       5;
[3]  105,      77,       7;
[4]  945,      1044,     234,      9;
[5]  10395,    14784,    5390,     550,      11;
[6]  135135,   227877,   113126,   19760,    1105,    13;
[7]  2027025,  3862305,  2371845,  586425,   58275,   1995,   15;
[8]  34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17;
[9]  ...
		

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.

Programs

Mathematica

max = 11; 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] // Simplify];
s = y0[x, t];
se = (1 + x*s + 2*x^2*D[s, x])/(1 - x*s)^2  + O[x]^max // Normal;
row[n_] := row[n] = CoefficientList[Coefficient[se, x, n], t];
T[0, 0] = 1; T[n_, k_] := row[n][[k + 1]];
Table[T[n, k], {n, 0, max-1}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)

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;
};
A286783_ser(N,t='t) = {
  my(s=A286781_ser(N,t)); (1 + x*s + 2*x^2*deriv(s,'x))/(1-x*s)^2;
};
concat(apply(p->Vecrev(p), Vec(A286783_ser(10))))

Formula

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

A001147(n+1)=T(n,0), A001700(n)=P_n(-1), A286794(n)=P_n(1).