%I #34 May 19 2017 04:07:45

%S 1,1,6,3,50,45,5,518,637,161,7,6354,9567,3744,414,9,89782,156123,

%T 80784,14850,880,11,1435330,2781389,1749969,446706,46150,1651,13,

%U 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

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

%C Row n>0 contains n terms.

%C 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).

%H Gheorghe Coserea, <a href="/A286782/b286782.txt">Rows n=0..123, flattened</a>

%H Luca G. Molinari, <a href="https://arxiv.org/abs/cond-mat/0401500">Hedin's equations and enumeration of Feynman's diagrams</a>, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

%F 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.

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

%e A(x;t) = 1 + x + (6 + 3*t)*x^2 + (50 + 45*t + 5*t^2)*x^3 + ...

%e Triangle starts:

%e n\k [0] [1] [2] [3] [4] [5] [6] [7]

%e [0] 1;

%e [1] 1;

%e [2] 6, 3;

%e [3] 50, 45, 5;

%e [4] 518, 637, 161, 7;

%e [5] 6354, 9567, 3744, 414, 9;

%e [6] 89782, 156123, 80784, 14850, 880, 11;

%e [7] 1435330, 2781389, 1749969, 446706, 46150, 1651, 13;

%e [8] 25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835, 15;

%e [9] ...

%t 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]];

%t row[n_] := (2n+1) CoefficientList[Coefficient[y0[x, t], x, n], t];

%t T[0, 0] = 1; T[n_, k_] := row[n-1][[k+1]];

%t 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 *)

%o (PARI)

%o A286781_ser(N,t='t) = {

%o my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);

%o while(n++,

%o y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;

%o if (y1 == y0, break()); y0 = y1;);

%o y0;

%o };

%o A286782_ser(N,t='t) = my(s=A286781_ser(N,t)); 1 + x*s + 2*x^2 * deriv(s,'x);

%o concat(apply(p->Vecrev(p), Vec(A286782_ser(10))))

%K nonn,tabf

%O 0,3

%A _Gheorghe Coserea_, May 14 2017