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A286800 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 4

%I

%S 1,1,2,7,6,63,74,10,729,974,254,8,10113,15084,5376,406,161935,264724,

%T 117424,14954,320,2923135,5163276,2697804,481222,23670,112,58547761,

%U 110483028,65662932,14892090,1186362,21936,1286468225,2570021310,1695874928,461501018,51034896,1866986,11264,30747331223,64547199082,46461697760,14603254902,2055851560,116329886,1905888,2560

%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="/A286800/b286800.txt">Rows n=1..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 x*deriv(y,x) = (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y), with y(0;t) = 0, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.

%F A049464(n) = T(n,0), P_n(-1) = (-1)^(n-1), A287029(n) = P_n(1).

%e A(x;t) = x + (1 + 2*t)*x^2 + (7 + 6*t)*x^3 + (63 + 74*t + 10*t^2)*x^4 + ...

%e Triangle starts:

%e n\k [0] [1] [2] [3] [4] [5]

%e [1] 1;

%e [2] 1, 2;

%e [3] 7, 6;

%e [4] 63, 74, 10;

%e [5] 729, 974, 254, 8;

%e [6] 10113, 15084, 5376, 406;

%e [7] 161935, 264724, 117424, 14954, 320;

%e [8] 2923135, 5163276, 2697804, 481222, 23670, 112;

%e [9] 58547761, 110483028, 65662932, 14892090, 1186362, 21936;

%e [10] ...

%t max = 12; y0[0, _] = y1[0, _] = 0; y0[x_, t_] = x; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1/(-1 + y0[x, t]))*x*(-1 - y0[x, t]^2 - 2*y0[x, t]*(-1 + D[y0[x, t], x]) + t*x*(-1 + y0[x, t])*(2*(-1 + y0[x, t])^2 + (x*(-1 + y0[x, t]) + y0[x, t])*D[y0[x, t], x])) + O[x]^n]; y0[x_, t_] = y1[x, t]];

%t row[n_] := CoefficientList[SeriesCoefficient[y0[x, t], {x, 0, n}], t];

%t Flatten[Table[row[n], {n, 0, max-1}]] (* _Jean-Fran├žois Alcover_, May 24 2017, adapted from PARI *)

%o (PARI)

%o A286795_ser(N, t='t) = {

%o my(x='x+O('x^N), y0=1, y1=0, n=1);

%o while(n++,

%o y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');

%o y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;

%o };

%o A286798_ser(N,t='t) = {

%o my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));

%o };

%o A286800_ser(N, t='t) = {

%o my(v = A286798_ser(N,t)); 1-1/subst(v, 'x, serreverse(x*v^2));

%o };

%o concat(apply(p->Vecrev(p), Vec(A286800_ser(12))))

%o \\ test: y=A286800_ser(50); x*y' == (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y)

%Y Cf. A286781, A286782, A286783, A286784, A286785.

%K nonn,tabf

%O 1,3

%A _Gheorghe Coserea_, May 22 2017

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Last modified July 24 03:01 EDT 2019. Contains 325290 sequences. (Running on oeis4.)