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.

A281270 a(n) is the number of closed BCK (a.k.a. affine) lambda terms of size n.

Original entry on oeis.org

0, 0, 1, 2, 3, 9, 30, 81, 242, 838, 2799, 9365, 33616, 122937, 449698, 1696724, 6558855, 25559806, 101294687, 409363758, 1673735259, 6928460475, 29115833976, 123835124242, 532449210893, 2317382872404, 10199542298725, 45345006540851, 203704505953902, 924427259637953, 4234544300812834
Offset: 0

Views

Author

Gheorghe Coserea, Apr 02 2017

Keywords

Comments

It appears that for n >= 1, a(n + 5) == a(n) (mod 5), a(n + 38*7) == a(n) (mod 7), a(n + 30*11) == a(n) (mod 11) and a(n + 288*17) == a(n) (mod 17). - Peter Bala, Apr 11 2022

Examples

			A(x) = x^2 + 2*x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 81*x^7 + 242*x^8 + ...

Links

Crossrefs

Cf. A062980, A262301, A267827.

Programs

  • Mathematica
    a[0] = a[1] = 0; a[n_] := a[n] = 1 + a[n - 1] + 2 Sum[ k a[k], {k, 2, n - 3}] + Sum[a[k] a[n - 1 - k], {k, 2, n - 3}]; Table[a@ n, {n, 0, 30}] (* Michael De Vlieger, Apr 02 2017 *)
  • PARI
    seq(N) = {
  my(a = vector(N));
  for (n=2, N, my(s1 = sum(k=2, n-3, k*a[k]));
    a[n] = 1 + a[n-1] + 2*s1 + sum(k=2, n-3, a[k]*a[n-1-k]));
  concat(0,a);
};
seq(30)
\\ test: y = Ser(seq(201)); 0 == 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2

Formula

a(n) = 1 + a(n-1) + 2*Sum_{k=2..n-3} k*a(k) + Sum_{k=2..n-3} a(k)*a(n-1-k) for n>=2.
0 = 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2, where y(x) is the g.f.
a(3*n+1) = Sum_{k=0..n-1} binomial(3*n,3*k+1)*A062980(k).

Extensions

Name clarified by Pierre Lescanne, May 19 2017

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.

A318110 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, 3, 3, 1, 26, 26, 11, 2, 367, 367, 167, 42, 5, 7142, 7142, 3352, 944, 163, 14, 176766, 176766, 84308, 25006, 4965, 638, 42, 5304356, 5304356, 2554329, 779246, 165474, 24924, 2510, 132, 186954535, 186954535, 90600599, 28120586, 6200455, 1010814, 121086, 9908, 429, 7566084686, 7566084686, 3683084984, 1156456088, 261067596, 44535120, 5829880, 574128, 39203, 1430
Offset: 0

Views

Author

Gheorghe Coserea, Sep 05 2018

Keywords

Examples

			A(x,t) = (1+t)*x + (3+3*t+t^2)*x^2 + (26+26*t+11*t^2+2*t^3)*x^3 + ...
Triangle starts:
n\k [0]       [1]       [2]      [3]      [4]     [5]     [6]    [7]  [8]
[0] 0;
[1] 1,        1;
[2] 3,        3,        1;
[3] 26,       26,       11,      2;
[4] 367,      367,      167,     42,      5;
[5] 7142,     7142,     3352,    944,     163,    14;
[6] 176766,   176766,   84308,   25006,   4965,   638,    42;
[7] 5304356,  5304356,  2554329, 779246,  165474, 24924,  2510,  132;
[8] 186954535,186954535,90600599,28120586,6200455,1010814,121086,9908,429;
[9] ...

Links

Crossrefs

Column 0 gives A262301.
Main diagonal gives A000108(n-1) for n>0.
Second diagonal gives A032443(n-1) for n>0.

Programs

  • Mathematica
    rows = 10; Clear[A]; A[x_, t_] = (1+t)x;
Do[A[x_, t_] = Series[x t/(1-A[x, t]) + D[A[x, t], t], {x, 0, n}, {t, 0, n}] // Normal, {n, 2 rows}];
CoefficientList[#, t]& /@ CoefficientList[A[x, t], x] /. {} -> {0} // Take[#, rows]& // Flatten (* Jean-François Alcover, Oct 23 2018 *)
  • PARI
    seq(N) = {
  my(x='x+O('x^N), t='t, F0=(1+t)*x, F1=0, n=1);
  while(n++,
    F1 = F0^2; F1 = F1 - deriv(F1,'t)/2 + deriv(F0,'t) + x*t;
    if (F1 == F0, break()); F0 = F1);
  concat([[0]], apply(Vecrev, Vec(F0)));
};
concat(seq(10))
\\ test: y=Ser(apply(p->Polrev(p,'t), seq(101)), 'x); y == x*'t/(1-y) + deriv(y,'t)

Formula

A(x,t) = Sum_{n>=0} P_n(t)*x^n, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k, satisfies:
A = x*t/(1-A) + deriv(A,t), with A(0,t) = 0, deriv(A,x)(0,t) = 1+t (deriv(A,v) represents the derivative of A with respect to variable v).
Showing 1-5 of 5 results.

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