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

1, 0, 1, 5, 3, 36, 33, 2, 329, 388, 72, 3655, 5101, 1545, 64, 47844, 75444, 30700, 3023, 20, 721315, 1248911, 621937, 97200, 3134, 12310199, 22964112, 13269140, 2793713, 180936, 1656, 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352, 4939227215, 10316541393, 7336995966, 2239771686, 293933437, 13977294, 140660

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)

A278990(n) = P_n(0), A294166(n) = P_n(1), A082582(n) = P_n(-1) for n > 1.

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

Cf. A082582, A267827, A278990, A286795, A286798, A286800.

Sequence in context: A189747 A279066 A179210 * A187278 A288184 A323779

Adjacent sequences: A291840 A291841 A291842 * A291844 A291845 A291846

Keyword

nonn,tabf

Author

Gheorghe Coserea, Oct 23 2017

Status

approved