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A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286795 #22 May 26 2017 22:18:00
%S A286795 1,1,4,3,27,31,5,248,357,117,7,2830,4742,2218,314,9,38232,71698,42046,
%T A286795 9258,690,11,593859,1216251,837639,243987,30057,1329,13,10401712,
%U A286795 22877725,17798029,6314177,1071809,81963,2331,15,202601898,472751962,404979234,166620434,35456432,3857904,196532,3812,17,4342263000,10651493718,9869474106,4561150162,1149976242,160594860,11946360,426852,5904,19
%N A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286795 Row n>0 contains n terms.
%C A286795 "The series expansion of the solution counts skeleton vertex diagrams with dressed propagators and bare interactions." (see G^2v-skeleton expansion in Molinari link)
%H A286795 Gheorghe Coserea, <a href="/A286795/b286795.txt">Rows n=0..123, flattened</a>
%H A286795 Luca G. Molinari, Nicola Manini, <a href="https://arxiv.org/abs/cond-mat/0512342">Enumeration of many-body skeleton diagrams</a>, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
%F A286795 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 0 = 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*deriv(y,x), with y(0;t)=1, where P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
%F A286795 A000699(n+1) = T(n,0), 1 = P_n(-1), A049464(n+1) = P_n(1).
%e A286795 A(x;t) = 1 + x + (4 + 3*t)*x^2 + (27 + 31*t + 5*t^2)*x^3 + ...
%e A286795 Triangle starts:
%e A286795 n\k  [0]       [1]       [2]       [3]      [4]      [5]    [6]   [7]
%e A286795 [0]  1;
%e A286795 [1]  1;
%e A286795 [2]  4,        3;
%e A286795 [3]  27,       31,       5;
%e A286795 [4]  248,      357,      117,      7;
%e A286795 [5]  2830,     4742,     2218,     314,     9;
%e A286795 [6]  38232,    71698,    42046,    9258,    690,     11;
%e A286795 [7]  593859,   1216251,  837639,   243987,  30057,   1329,  13;
%e A286795 [8]  10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15;
%e A286795 [9] ...
%t A286795 max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = ((1 + x*(1 + 2 t + x t^2) y0[x, t]^2 + t (1 - t)*x^2*y0[x, t]^3 + 2 x^2 y0[x, t] D[y0[x, t], x]))/(1 + 2 x*t) + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
%t A286795 row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
%t A286795 Table[row[n], {n, 0, max - 1}] // Flatten (* _Jean-François Alcover_, May 23 2017, adapted from PARI *)
%o A286795 (PARI)
%o A286795 A286795_ser(N, t='t) = {
%o A286795   my(x='x+O('x^N), y0=1, y1=0, n=1);
%o A286795   while(n++,
%o A286795     y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
%o A286795     y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
%o A286795 };
%o A286795 concat(apply(p->Vecrev(p), Vec(A286795_ser(11))))
%o A286795 \\ test: y=A286795_ser(50); 0 == 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*y'
%Y A286795 Cf. A286781, A286782, A286783, A286784, A286785.
%K A286795 nonn,tabf
%O A286795 0,3
%A A286795 _Gheorghe Coserea_, May 21 2017