OFFSET
0,4
COMMENTS
Row n > 0 contains floor((2*n+1)/3) terms.
LINKS
Gheorghe Coserea, Rows n = 0..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 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)
A267827(n) = T(3*n+1, 2*n), n > 0. - Danny Rorabaugh, Nov 10 2017
EXAMPLE
A(x;t) = 1 + x^2 + (5 + 3*t)*x^3 + (36 + 33*t + 2*t^2)*x^4 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6]
[0] 1;
[1] 0;
[2] 1;
[3] 5, 3;
[4] 36, 33, 2;
[5] 329, 388, 72;
[6] 3655, 5101, 1545, 64;
[7] 47844, 75444, 30700, 3023, 20;
[8] 721315, 1248911, 621937, 97200, 3134;
[9] 12310199, 22964112, 13269140, 2793713, 180936, 1656;
[10] 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352;
[11] ...
MATHEMATICA
nmax = 11; Clear[Z, Zp]; Z[_] = 0;
Do[
Zp[t_] = Z'[t] + O[t]^n // Normal;
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,
{n, nmax+1}];
CoefficientList[#, L]& /@ CoefficientList[Z[t], t] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 23 2018 *)
PROG
(PARI)
A291843_ser(N, t='t) = {
my(x='x+O('x^N), y=1, y1=0, n=1,
dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
while (n++,
y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
(t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
if (y1 == y, break); y = y1; ); y;
};
concat(apply(p->if(p === Pol(0, 't), [0], Vecrev(p)), Vec(A291843_ser(12))))
\\ 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)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, Oct 23 2017
STATUS
approved