A287040 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

0, 1, 1, 1, 1, 2, 5, 3, 2, 8, 17, 22, 10, 5, 29, 91, 106, 94, 35, 14, 140, 431, 701, 582, 396, 126, 42, 661, 2501, 4067, 4544, 2980, 1654, 462, 132, 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429, 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430, 120909, 550811, 1245517, 1727148, 1776310, 1291654, 772422, 322204, 117016, 24310, 4862

Offset

0,6

Links

Gheorghe Coserea, Rows n=0..200, flattened

Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017.

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = t + x*y^2 + x*deriv(y,t) + x*y, with y(0;t)=t, where P_n(t) = Sum_{k=0..n+1} T(n,k)*t^k.

A000108(n)=T(n,n+1), A001700(n)=T(n+1,n+1).

Example

A(x;t) = t + (1 + t + t^2)*x + (2 + 5*t + 3*t^2 + 2*t^3)*x^2 + (8 + 17*t + 22*t^2 + 10*t^3 + 5*t^4)*x^3 + ...
Triangle starts:
n\k [0]    [1]    [2]     [3]     [4]     [5]     [6]    [7]    [8]   [9]
[0] 0,     1;
[1] 1,     1,     1;
[2] 2,     5,     3,      2;
[3] 8,     17,    22,     10,     5;
[4] 29,    91,    106,    94,     35,     14;
[5] 140,   431,   701,    582,    396,    126,    42;
[6] 661,   2501,  4067,   4544,   2980,   1654,   462,   132;
[7] 3622,  14025, 27394,  31032,  26680,  14598,  6868,  1716,  429;
[8] 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430;
[9] ...

Mathematica

nmax = 10; y[0, t_] := t; y[_, _] = 0;
Do[y[x_, t_] = Series[t + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, nmax}, {t, 0, nmax}] // Normal, {n, 0, nmax}];
CoefficientList[#, t]& /@ CoefficientList[y[x, t]+O[x]^nmax, x] // Flatten (* Jean-François Alcover, Dec 13 2018 *)

Prog

(PARI)
A287040_ser(N) = {
  my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
  while(n++,
    F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
    if (F1 == F0, break()); F0 = F1; ); F0;
};
concat(apply(p->Vecrev(p), Vec(A287040_ser(10))))
\\ test: y=A287040_ser(50); y == t + x*y^2 + x*deriv(y, t) + x*y

Crossrefs

Cf. A262301, A267827, A281270, A287030, A287045 (column 0).

Sequence in context: A156637 A195784 A279623 * A212887 A163766 A004200

Adjacent sequences: A287037 A287038 A287039 * A287041 A287042 A287043

Keyword

nonn,tabl

Author

Gheorghe Coserea, May 23 2017

Status

approved