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

%I A286783 #29 May 19 2017 11:41:06
%S A286783 1,3,15,5,105,77,7,945,1044,234,9,10395,14784,5390,550,11,135135,
%T A286783 227877,113126,19760,1105,13,2027025,3862305,2371845,586425,58275,
%U A286783 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 A286783 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286783 Row n>0 contains n terms.
%C A286783 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 A286783 Gheorghe Coserea, <a href="/A286783/b286783.txt">Rows n=0..123, flattened</a>
%H A286783 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 A286783 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 A286783 A001147(n+1)=T(n,0), A001700(n)=P_n(-1), A286794(n)=P_n(1).
%e A286783 A(x;t) = 1 + 3*x + (15 + 5*t)*x^2 + (105 + 77*t + 7*t^2)*x^3 + ...
%e A286783 Triangle starts:
%e A286783 n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]   [7]
%e A286783 [0]  1;
%e A286783 [1]  3;
%e A286783 [2]  15,       5;
%e A286783 [3]  105,      77,       7;
%e A286783 [4]  945,      1044,     234,      9;
%e A286783 [5]  10395,    14784,    5390,     550,      11;
%e A286783 [6]  135135,   227877,   113126,   19760,    1105,    13;
%e A286783 [7]  2027025,  3862305,  2371845,  586425,   58275,   1995,   15;
%e A286783 [8]  34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17;
%e A286783 [9]  ...
%t A286783 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 A286783 s = y0[x, t];
%t A286783 se = (1 + x*s + 2*x^2*D[s, x])/(1 - x*s)^2  + O[x]^max // Normal;
%t A286783 row[n_] := row[n] = CoefficientList[Coefficient[se, x, n], t];
%t A286783 T[0, 0] = 1; T[n_, k_] := row[n][[k + 1]];
%t A286783 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 A286783 (PARI)
%o A286783 A286781_ser(N,t='t) = {
%o A286783   my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
%o A286783   while(n++,
%o A286783     y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
%o A286783     if (y1 == y0, break()); y0 = y1;);
%o A286783   y0;
%o A286783 };
%o A286783 A286783_ser(N,t='t) = {
%o A286783   my(s=A286781_ser(N,t)); (1 + x*s + 2*x^2*deriv(s,'x))/(1-x*s)^2;
%o A286783 };
%o A286783 concat(apply(p->Vecrev(p), Vec(A286783_ser(10))))
%K A286783 nonn,tabf
%O A286783 0,2
%A A286783 _Gheorghe Coserea_, May 14 2017