%I #29 May 19 2017 11:41:06

%S 1,3,15,5,105,77,7,945,1044,234,9,10395,14784,5390,550,11,135135,

%T 227877,113126,19760,1105,13,2027025,3862305,2371845,586425,58275,

%U 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

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

%H Gheorghe Coserea, <a href="/A286783/b286783.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))/(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.

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

%e A(x;t) = 1 + 3*x + (15 + 5*t)*x^2 + (105 + 77*t + 7*t^2)*x^3 + ...

%e Triangle starts:

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

%e [0] 1;

%e [1] 3;

%e [2] 15, 5;

%e [3] 105, 77, 7;

%e [4] 945, 1044, 234, 9;

%e [5] 10395, 14784, 5390, 550, 11;

%e [6] 135135, 227877, 113126, 19760, 1105, 13;

%e [7] 2027025, 3862305, 2371845, 586425, 58275, 1995, 15;

%e [8] 34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17;

%e [9] ...

%t 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];

%t s = y0[x, t];

%t se = (1 + x*s + 2*x^2*D[s, x])/(1 - x*s)^2 + O[x]^max // Normal;

%t row[n_] := row[n] = CoefficientList[Coefficient[se, x, n], t];

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

%t 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 *)

%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 A286783_ser(N,t='t) = {

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

%o };

%o concat(apply(p->Vecrev(p), Vec(A286783_ser(10))))

%K nonn,tabf

%O 0,2

%A _Gheorghe Coserea_, May 14 2017