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A291843 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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