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A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 2

%I #18 May 26 2017 22:18:09

%S 1,1,1,4,5,1,27,40,14,1,248,419,200,30,1,2830,5308,3124,700,55,1,

%T 38232,78070,53620,15652,1960,91,1,593859,1301088,1007292,356048,

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

%H Gheorghe Coserea, <a href="/A286796/b286796.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 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 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 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 Triangle starts:

%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8]

%e [0] 1;

%e [1] 1; 1;

%e [2] 4, 5, 1;

%e [3] 27, 40, 14, 1;

%e [4] 248, 419, 200, 30, 1;

%e [5] 2830, 5308, 3124, 700, 55, 1;

%e [6] 38232, 78070, 53620, 15652, 1960, 91, 1;

%e [7] 593859, 1301088, 1007292, 356048, 60550, 4704, 140, 1;

%e [8] 10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1;

%e [9] ...

%t 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 row[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t];

%t Flatten[Table[row[n], {n, 0, max-1}]] (* _Jean-François Alcover_, May 23 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 A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);

%o concat(apply(p->Vecrev(p), Vec(A286796_ser(11))))

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

%K nonn,tabl

%O 0,4

%A _Gheorghe Coserea_, May 21 2017

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)