A287040 -id:A287040 - cp's OEIS frontend

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

Gheorghe Coserea, May 23 2017

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

			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]  ...

Gheorghe Coserea, Rows n=0..200, flattened
Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017.

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

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 *)

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

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).

A287045 a(n) is the number of size n affine closed terms of variable size 0.

Original entry on oeis.org

0, 1, 2, 8, 29, 140, 661, 3622, 19993, 120909, 744890, 4887401, 32795272, 230728608, 1661537689, 12426619200, 95087157771, 750968991327, 6062088334528, 50288003979444, 425889463252945, 3694698371069796, 32683415513480237, 295430131502604353, 2719833636188015674, 25536232370225996575

Offset: 0

Gheorghe Coserea, May 28 2017

nonn

			A(x) = x + 2*x^2 + 8*x^3 + 29*x^4 + 140*x^5 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017.

Column zero of A287040.

Cf. A262301, A267827, A281270, A287030.

a[n_] := a[n] = If[n<3, n, (3a[n-1] + (6n-10) a[n-2] - a[n-3] + 2b[n-1] - b[n-2] - b[n-3])/2]; b[n_] := Sum[a[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 13 2018 *)

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(0, Vec(subst(A287040_ser(26), 't, 0)))

A287045_seq(N) = {
  my(a = vector(N), b=vector(N), t1=0);
  a[1]=1; a[2]=2; a[3]=8; b[1]=0; b[2]=1; b[3]=4;
  for (n=4, N, b[n] = sum(k=1, n-1, a[k]*a[n-k]);
    t1 = 3*a[n-1] + (6*n-10)*a[n-2] - a[n-3];
    a[n] = (t1 + 2*b[n-1] - b[n-2] - b[n-3])/2);
  concat(0,a);
};
A287045_seq(25)
\\ test: y=Ser(A287045_seq(200)); 0 == 6*x^3*y' - x*(x-1)*(x+2)*y^2 - (x^3-2*x^2-3*x+2)*y + x^2 + 2*x

A(x) = A287040(x;0).
a(n) = (3*a(n-1) + (6*n-10)*a(n-2) - a(n-3) + 2*b(n-1) - b(n-2) - b(n-3))/2, where b(n) = Sum_{k=1..n-1} a(k)*a(n-k).
0 = 6*x^3*deriv(y,x) - x*(x-1)*(x+2)*y^2 - (x^3-2*x^2-3*x+2)*y + x^2 + 2*x, where y(x) is the g.f.

cp's OEIS Frontend

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

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A287045 a(n) is the number of size n affine closed terms of variable size 0.

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