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A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286796 #18 May 26 2017 22:18:09
%S A286796 1,1,1,4,5,1,27,40,14,1,248,419,200,30,1,2830,5308,3124,700,55,1,
%T A286796 38232,78070,53620,15652,1960,91,1,593859,1301088,1007292,356048,
%U A286796 60550,4704,140,1,10401712,24177939,20604768,8430844,1787280,194854,10080,204,1,202601898,495263284,456715752,209878440,52619854,7322172,545908,19800,285,1,4342263000,11085720018,10921213644,5516785032,1579263840,264576774,25677652,1372228,36300,385,1
%N A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%H A286796 Gheorghe Coserea, <a href="/A286796/b286796.txt">Rows n=0..123, flattened</a>
%H A286796 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 A286796 A(x;t) = Sum_{n>=0} P_n(t)*x^n = v/(1-x*t*v), where v(x;t) = A286795(x;t) and P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
%F A286796 A000699(n+1)=T(n,0), A000330(n)=T(n,n-1), A286797(n)=P_n(1) and P_n(-1)=0 for n>0.
%e A286796 A(x;t) = 1 + (1 + t)*x + (4 + 5*t + t^2)*x^2 + (27 + 40*t + 14*t^2 + t^3)*x^3 + ...
%e A286796 Triangle starts:
%e A286796 n\k  [0]       [1]       [2]       [3]      [4]      [5]     [6]    [7]  [8]
%e A286796 [0]  1;
%e A286796 [1]  1;        1;
%e A286796 [2]  4,        5,        1;
%e A286796 [3]  27,       40,       14,       1;
%e A286796 [4]  248,      419,      200,      30,      1;
%e A286796 [5]  2830,     5308,     3124,     700,     55,      1;
%e A286796 [6]  38232,    78070,    53620,    15652,   1960,    91,     1;
%e A286796 [7]  593859,   1301088,  1007292,  356048,  60550,   4704,   140,   1;
%e A286796 [8]  10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1;
%e A286796 [9]  ...
%t A286796 max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(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]; y0[x_, t_] = y1[x, t]];
%t A286796 row[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t];
%t A286796 Flatten[Table[row[n], {n, 0, max-1}]] (* _Jean-François Alcover_, May 23 2017, adapted from PARI *)
%o A286796 (PARI)
%o A286796 A286795_ser(N, t='t) = {
%o A286796   my(x='x+O('x^N), y0=1, y1=0, n=1);
%o A286796   while(n++,
%o A286796     y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
%o A286796     y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
%o A286796 };
%o A286796 A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);
%o A286796 concat(apply(p->Vecrev(p), Vec(A286796_ser(11))))
%Y A286796 Cf. A286781, A286782, A286783, A286784, A286785.
%K A286796 nonn,tabl
%O A286796 0,4
%A A286796 _Gheorghe Coserea_, May 21 2017