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A286798 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. .

%I A286798 #19 May 26 2017 22:18:24
%S A286798 1,1,4,2,27,22,248,264,30,2830,3610,830,8,38232,55768,18746,1078,
%T A286798 593859,961740,414720,46986,576,10401712,18326976,9457788,1593664,
%U A286798 62682,112,202601898,382706674,226526362,49941310,3569882,45296,4342263000,8697475368,5740088706,1540965514,160998750,4909674,16896,101551822350,213865372020,154271354280,48205014786,6580808784,337737294,4200032,2560
%N A286798 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. .
%C A286798 Row n>0 contains floor(2*(n+1)/3) terms.
%H A286798 Gheorghe Coserea, <a href="/A286798/b286798.txt">Rows n=0..123, flattened</a>
%H A286798 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 A286798 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies x^2*deriv(y,x) = (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2), with y(0;t) = 1, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.
%F A286798 A000699(n+1)=T(n,0), A000108(n)=P_n(-1), A286799(n)=P_n(1).
%e A286798 A(x;t) = 1 + x + (4 + 2*t)*x^2 + (27 + 22*t)*x^3 + (248 + 264*t + 30*t^2)*x^4 +
%e A286798 Triangle starts:
%e A286798 n\k  [0]        [1]        [2]        [3]       [4]      [5]
%e A286798 [0]  1;
%e A286798 [1]  1;
%e A286798 [2]  4,         2;
%e A286798 [3]  27,        22;
%e A286798 [4]  248,       264,       30;
%e A286798 [5]  2830,      3610,      830,       8;
%e A286798 [6]  38232,     55768,     18746,     1078;
%e A286798 [7]  593859,    961740,    414720,    46986,    576;
%e A286798 [8]  10401712,  18326976,  9457788,   1593664,  62682,   112;
%e A286798 [9]  202601898, 382706674, 226526362, 49941310, 3569882, 45296;
%e A286798 [10] ...
%t A286798 max = 12; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = 1 + x y0[x, t]^2 + 3 t x^3 y0[x, t]^2 D[y0[x, t], x] + x^2 (2 y0[x, t] D[y0[x, t], x] + t (2 y0[x, t]^3 - D[y0[x, t], x] + y0[x, t] D[y0[x, t], x])) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];
%t A286798 P[n_, t_] := Coefficient[y0[x, t] , x, n];
%t A286798 row[n_] := CoefficientList[P[n, t], t];
%t A286798 Table[row[n], {n, 0, max}] // Flatten (* _Jean-François Alcover_, May 24 2017, adapted from PARI *)
%o A286798 (PARI)
%o A286798 A286795_ser(N, t='t) = {
%o A286798   my(x='x+O('x^N), y0=1, y1=0, n=1);
%o A286798   while(n++,
%o A286798     y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
%o A286798     y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
%o A286798 };
%o A286798 A286798_ser(N,t='t) = {
%o A286798   my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
%o A286798 };
%o A286798 concat(apply(p->Vecrev(p), Vec(A286798_ser(12))))
%o A286798 \\ test: y=A286798_ser(50); x^2*y' == (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2)
%Y A286798 Cf. A286781, A286782, A286783, A286784, A286785.
%K A286798 nonn,tabf
%O A286798 0,3
%A A286798 _Gheorghe Coserea_, May 21 2017