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A286781 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286781 #42 May 19 2017 04:07:49
%S A286781 1,2,1,10,9,1,74,91,23,1,706,1063,416,46,1,8162,14193,7344,1350,80,1,
%T A286781 110410,213953,134613,34362,3550,127,1,1708394,3602891,2620379,842751,
%U A286781 125195,8085,189,1,29752066,67168527,54636792,20862684,4009832,382358,16576,268,1,576037442,1375636129,1223392968,533394516,124266346,15653598,1023340,31356,366,1
%N A286781 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286781 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 self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).
%H A286781 Gheorghe Coserea, <a href="/A286781/b286781.txt">Rows n=0..122, flattened</a>
%H A286781 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 A286781 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y * (1-x*y)^2 = (1 + x*y + 2*x^2*deriv(y,x)) * (1 - x*y*(1-t)), with y(0;t) = 1, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k, 0<=n, 0<=k<=n.
%F A286781 A000698(n+1)=T(n,0), A101986(n)=T(n,n-1), A000108(n)=P_n(-1), A286794(n)=P_n(1).
%e A286781 A(x;t) = 1 + (2 + t)*x + (10 + 9*t + t^2)*x^2 + (74 + 91*t + 23*t^2 + t^3)*x^3 + ...
%e A286781 Triangle starts:
%e A286781 n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]  [8]
%e A286781 [0]  1;
%e A286781 [1]  2,        1;
%e A286781 [2]  10,       9,        1;
%e A286781 [3]  74,       91,       23,       1;
%e A286781 [4]  706,      1063,     416,      46,       1;
%e A286781 [5]  8162,     14193,    7344,     1350,     80,      1;
%e A286781 [6]  110410,   213953,   134613,   34362,    3550,    127,    1;
%e A286781 [7]  1708394,  3602891,  2620379,  842751,   125195,  8085,   189,   1;
%e A286781 [8]  29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1;
%e A286781 [9] ...
%t A286781 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 A286781 row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
%t A286781 Table[row[n], {n, 0, max-1}] // Flatten (* _Jean-François Alcover_, May 19 2017, adapted from PARI *)
%o A286781 (PARI)
%o A286781 A286781_ser(N,t='t) = {
%o A286781   my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
%o A286781   while(n++,
%o A286781     y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
%o A286781     if (y1 == y0, break()); y0 = y1;);
%o A286781   y0;
%o A286781 };
%o A286781 concat(apply(p->Vecrev(p), Vec(A286781_ser(10))))
%o A286781 \\ test: y = A286781_ser(50); y*(1-x*y)^2 == (1 + x*y + 2*x^2*deriv(y,'x)) * (1 - x*y*(1-t))
%Y A286781 For vertex and polarization functions see A286782 and A286783.  For GWA of the self-energy and polarization functions see A286784 and A286785.
%Y A286781 Columns k=0-8 give: A000698(k=0), A286786(k=1), A286787(k=2), A286788(k=3), A286789(k=4), A286790(k=5), A286791(k=6), A286792(k=7), A286793(k=8).
%K A286781 nonn,tabl
%O A286781 0,2
%A A286781 _Gheorghe Coserea_, May 14 2017