cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Gheorghe Coserea, May 23 2017

Keywords

Comments

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

Examples

			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]  ...

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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).

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

Original entry on oeis.org

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

Views

Author

Gheorghe Coserea, May 23 2017

Keywords

Examples

			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] ...

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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).

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

Views

Author

Gheorghe Coserea, May 28 2017

Keywords

Examples

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

Links

Crossrefs

Column zero of A287040.
Cf. A262301, A267827, A281270, A287030.

Programs

  • Mathematica
    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 *)
  • 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(0, Vec(subst(A287040_ser(26), 't, 0)))
  • PARI
    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

Formula

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.

A287141 Numbers of closed affine (a.k.a. BCK) lambda terms of natural size n.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 12, 25, 64, 166, 405, 1050, 2763, 7239, 19190, 51457, 138538, 374972, 1020943, 2792183, 7666358, 21126905, 58422650, 162052566, 450742451, 1256974690, 3513731861, 9843728012, 27633400879, 77721141911, 218984204904, 618021576627, 1746906189740, 4945026080426, 14017220713131
Offset: 0

Views

Author

Pierre Lescanne, May 20 2017

Keywords

Links

Crossrefs

Cf. A281270, A275057.

A359181 Number of commutative BCK-algebras of order n up to isomorphism.

Original entry on oeis.org

1, 2, 5, 11, 28, 72, 192, 515, 1426
Offset: 2

Views

Author

Choiwah Chow, Dec 18 2022

Keywords

Comments

a(2)-a(10) were generated using the model enumerator Mace4.

Links

Crossrefs

Cf. A287141, A062980, A281270.
Showing 1-5 of 5 results.

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