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%I #32 Oct 23 2018 08:22:53
%S 1,0,1,5,3,36,33,2,329,388,72,3655,5101,1545,64,47844,75444,30700,
%T 3023,20,721315,1248911,621937,97200,3134,12310199,22964112,13269140,
%U 2793713,180936,1656,234615096,465344235,301698501,78495574,7733807,205620,352,4939227215,10316541393,7336995966,2239771686,293933437,13977294,140660
%N Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C Row n > 0 contains floor((2*n+1)/3) terms.
%H Gheorghe Coserea, <a href="/A291843/b291843.txt">Rows n = 0..123, flattened</a>
%H 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 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 2*x^2*deriv(y,x) = (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x), with y(0;t)=1, where P_n(t) = Sum_{k=0..floor((2*n-2)/3)} T(n,k)*t^k for n > 0. (see eqn. (24) in Molinari link)
%F A278990(n) = P_n(0), A294166(n) = P_n(1), A082582(n) = P_n(-1) for n > 1.
%F A267827(n) = T(3*n+1, 2*n), n > 0. - _Danny Rorabaugh_, Nov 10 2017
%e A(x;t) = 1 + x^2 + (5 + 3*t)*x^3 + (36 + 33*t + 2*t^2)*x^4 + ...
%e Triangle starts:
%e n\k [0] [1] [2] [3] [4] [5] [6]
%e [0] 1;
%e [1] 0;
%e [2] 1;
%e [3] 5, 3;
%e [4] 36, 33, 2;
%e [5] 329, 388, 72;
%e [6] 3655, 5101, 1545, 64;
%e [7] 47844, 75444, 30700, 3023, 20;
%e [8] 721315, 1248911, 621937, 97200, 3134;
%e [9] 12310199, 22964112, 13269140, 2793713, 180936, 1656;
%e [10] 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352;
%e [11] ...
%t nmax = 11; Clear[Z, Zp]; Z[_] = 0;
%t Do[
%t Zp[t_] = Z'[t] + O[t]^n // Normal;
%t Z[t_] = (-(1/(2L t (1+t)))) (-1 + t - 2L t + 2L^2 t^4 (1 + Zp[t]) + t^2 (1 + 2L + 2L Zp[t]) + L t^3 (3 + 2L + 2(1+L) Zp[t]) + Sqrt[4L t (1+t) (1 + L t)(-1 + t + 2L t^2 + 2(-1 + L) t^2 Zp[t]) + (-1 + t (1 + t + L (-2 + t (2 + t (3 + 2L (1+t))))) + 2L t^2 (1+t)(1 + L t) Zp[t])^2]) + O[t]^n // Normal // Simplify,
%t {n, nmax+1}];
%t CoefficientList[#, L]& /@ CoefficientList[Z[t], t] /. {} -> {0} // Flatten (* _Jean-François Alcover_, Oct 23 2018 *)
%o (PARI)
%o A291843_ser(N, t='t) = {
%o my(x='x+O('x^N), y=1, y1=0, n=1,
%o dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
%o while (n++,
%o y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
%o (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
%o if (y1 == y, break); y = y1;); y;
%o };
%o concat(apply(p->if(p === Pol(0,'t), [0], Vecrev(p)), Vec(A291843_ser(12))))
%o \\ test: y=A291843_ser(56); 2*x^2*deriv(y,x) == (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x)
%Y Cf. A082582, A267827, A278990, A286795, A286798, A286800.
%K nonn,tabf
%O 0,4
%A _Gheorghe Coserea_, Oct 23 2017
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