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A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286782 #34 May 19 2017 04:07:45
%S A286782 1,1,6,3,50,45,5,518,637,161,7,6354,9567,3744,414,9,89782,156123,
%T A286782 80784,14850,880,11,1435330,2781389,1749969,446706,46150,1651,13,
%U A286782 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 A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286782 Row n>0 contains n terms.
%C A286782 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 A286782 Gheorghe Coserea, <a href="/A286782/b286782.txt">Rows n=0..123, flattened</a>
%H A286782 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 A286782 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 A286782 T(n+1,k) = (2*n+1)*A286781(n,k), A005416(n)=T(n,0), A088218(n)=P_n(-1).
%e A286782 A(x;t) = 1 + x + (6 + 3*t)*x^2 + (50 + 45*t + 5*t^2)*x^3 + ...
%e A286782 Triangle starts:
%e A286782 n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]
%e A286782 [0]  1;
%e A286782 [1]  1;
%e A286782 [2]  6,        3;
%e A286782 [3]  50,       45,       5;
%e A286782 [4]  518,      637,      161,      7;
%e A286782 [5]  6354,     9567,     3744,     414,      9;
%e A286782 [6]  89782,    156123,   80784,    14850,    880,     11;
%e A286782 [7]  1435330,  2781389,  1749969,  446706,   46150,   1651,   13;
%e A286782 [8]  25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835,  15;
%e A286782 [9]  ...
%t A286782 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 A286782 row[n_] := (2n+1) CoefficientList[Coefficient[y0[x, t], x, n], t];
%t A286782 T[0, 0] = 1; T[n_, k_] := row[n-1][[k+1]];
%t A286782 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 A286782 (PARI)
%o A286782 A286781_ser(N,t='t) = {
%o A286782   my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
%o A286782   while(n++,
%o A286782     y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
%o A286782     if (y1 == y0, break()); y0 = y1;);
%o A286782   y0;
%o A286782 };
%o A286782 A286782_ser(N,t='t) = my(s=A286781_ser(N,t)); 1 + x*s + 2*x^2 * deriv(s,'x);
%o A286782 concat(apply(p->Vecrev(p), Vec(A286782_ser(10))))
%K A286782 nonn,tabf
%O A286782 0,3
%A A286782 _Gheorghe Coserea_, May 14 2017