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

0,2

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).

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))/(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).

Example

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]  ...

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 *)

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;
};
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))))

Crossrefs

Sequence in context: A302782 A088558 A212203 * A351697 A012881 A248031

Adjacent sequences: A286780 A286781 A286782 * A286784 A286785 A286786

Keyword

nonn,tabf

Author

Gheorghe Coserea, May 14 2017

Status

approved