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A286800
Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
4
1, 1, 2, 7, 6, 63, 74, 10, 729, 974, 254, 8, 10113, 15084, 5376, 406, 161935, 264724, 117424, 14954, 320, 2923135, 5163276, 2697804, 481222, 23670, 112, 58547761, 110483028, 65662932, 14892090, 1186362, 21936, 1286468225, 2570021310, 1695874928, 461501018, 51034896, 1866986, 11264, 30747331223, 64547199082, 46461697760, 14603254902, 2055851560, 116329886, 1905888, 2560
OFFSET
1,3
COMMENTS
Row n>0 contains floor(2*(n+1)/3) terms.
LINKS
Gheorghe Coserea, Rows n=1..123, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
FORMULA
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.
A049464(n) = T(n,0), P_n(-1) = (-1)^(n-1), A287029(n) = P_n(1).
EXAMPLE
A(x;t) = x + (1 + 2*t)*x^2 + (7 + 6*t)*x^3 + (63 + 74*t + 10*t^2)*x^4 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5]
[1] 1;
[2] 1, 2;
[3] 7, 6;
[4] 63, 74, 10;
[5] 729, 974, 254, 8;
[6] 10113, 15084, 5376, 406;
[7] 161935, 264724, 117424, 14954, 320;
[8] 2923135, 5163276, 2697804, 481222, 23670, 112;
[9] 58547761, 110483028, 65662932, 14892090, 1186362, 21936;
[10] ...
MATHEMATICA
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]];
row[n_] := CoefficientList[SeriesCoefficient[y0[x, t], {x, 0, n}], t];
Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 24 2017, adapted from PARI *)
PROG
(PARI)
A286795_ser(N, t='t) = {
my(x='x+O('x^N), y0=1, y1=0, n=1);
while(n++,
y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1; ); y0;
};
A286798_ser(N, t='t) = {
my(v = A286795_ser(N, t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
A286800_ser(N, t='t) = {
my(v = A286798_ser(N, t)); 1-1/subst(v, 'x, serreverse(x*v^2));
};
concat(apply(p->Vecrev(p), Vec(A286800_ser(12))))
\\ 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)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, May 22 2017
STATUS
approved