A287030 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

0, 0, 1, 1, 1, 2, 1, 1, 3, 5, 3, 9, 17, 6, 2, 30, 41, 26, 10, 81, 131, 111, 30, 5, 242, 491, 357, 134, 35, 838, 1625, 1274, 652, 140, 14, 2799, 5497, 5202, 2556, 676, 126, 9365, 20581, 19827, 10200, 3610, 630, 42, 33616, 76561, 74797, 44880, 16390, 3334, 462, 122937, 282591, 301188, 190278, 72490, 19218, 2772, 132, 449698, 1089375, 1219920, 788654, 341770, 97890, 16108, 1716, 1696724, 4285737, 4893603, 3398950, 1578577, 474838, 99386, 12012, 429

Offset

0,6

Comments

Row n contains floor((n+3)/2) terms.

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

A281270(n)=T(n,0), A000108(n)=T(2*n+1,n+1), A001700(n-1)=T(2*n,n).

Example

A(x;t) = t*x + (1 + t)*x^2 + (2 + t + t^2)*x^3 + (3 + 5*t + 3*t^2)*x^4 + (9 + 17*t + 6*t^2 + 2*t^3)*x^5 + ...
Triangle starts:
n\k   [0]     [1]      [2]      [3]     [4]     [5]    [6]    [7]
[0]   0;
[1]   0,      1;
[2]   1,      1;
[3]   2,      1,       1;
[4]   3,      5,       3;
[5]   9,      17,      6,       2;
[6]   30,     41,      26,      10;
[7]   81,     131,     111,     30,     5;
[8]   242,    491,     357,     134,    35;
[9]   838,    1625,    1274,    652,    140,    14;
[10]  2799,   5497,    5202,    2556,   676,    126;
[11]  9365,   20581,   19827,   10200,  3610,   630,   42;
[12]  33616,  76561,   74797,   44880,  16390,  3334,  462;
[13]  122937, 282591,  301188,  190278, 72490,  19218, 2772,  132;
[14]  449698, 1089375, 1219920, 788654, 341770, 97890, 16108, 1716;
[15]  ...

Mathematica

max = 15; y[_, _] = 0;
Do[y[x_, t_] = Series[t x + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, max}, {t, 0, max}] // Normal, max];
CoefficientList[#, t]& /@ CoefficientList[y[x, t], x] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 25 2018 *)

Prog

(PARI)
A287030_ser(N) = {
  my(x='x+O('x^N), F0=x, t='t, F1=0, n=1);
  while(n++,
    F1 = t*x + x*F0^2 + x*deriv(F0, t) + x*F0;
    if (F1 == F0, break()); F0 = F1; ); F0;
};
concat(0, concat(apply(p->Vecrev(p), Vec(A287030_ser(16)))))
\\ test: y=A287030_ser(100); y == t*x + x*y^2 + x*deriv(y, t) + x*y

Crossrefs

Cf. A262301, A267827, A281270, A287040, A287045.

Sequence in context: A060082 A102225 A183262 * A187066 A187065 A174620

Adjacent sequences: A287027 A287028 A287029 * A287031 A287032 A287033

Keyword

nonn,tabf

Author

Gheorghe Coserea, May 23 2017

Status

approved