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 1-10 of 13 results. Next

A293808 Number T(n,k) of multisets of exactly k nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 10, 7, 2, 1, 0, 26, 18, 7, 2, 1, 0, 76, 56, 22, 7, 2, 1, 0, 232, 168, 68, 22, 7, 2, 1, 0, 764, 543, 218, 73, 22, 7, 2, 1, 0, 2620, 1792, 721, 234, 73, 22, 7, 2, 1, 0, 9496, 6187, 2438, 791, 240, 73, 22, 7, 2, 1, 0, 35696, 22088, 8491, 2702, 811, 240, 73, 22, 7, 2, 1
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2017

Keywords

Examples

			T(0,0) = 1: {}.
T(3,1) = 4: {aaa}, {aab}, {aba}, {abc}.
T(3,2) = 2: {a,aa}, {a,ab}.
T(3,3) = 1: {a,a,a}.
T(4,2) = 7: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,aa}, {aa,ab}, {ab,ab}.
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    2,    1;
  0,    4,    2,    1;
  0,   10,    7,    2,   1;
  0,   26,   18,    7,   2,   1;
  0,   76,   56,   22,   7,   2,  1;
  0,  232,  168,   68,  22,   7,  2,  1;
  0,  764,  543,  218,  73,  22,  7,  2, 1;
  0, 2620, 1792,  721, 234,  73, 22,  7, 2, 1;
  0, 9496, 6187, 2438, 791, 240, 73, 22, 7, 2, 1;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000085 (for n>0), A294004, A294005, A294006, A294007, A294008, A294009, A294010, A294011, A294012.
Row sums give: A293110.
T(2n,n) gives A293809.
Cf. A293815.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..15);
  • Mathematica
    g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]] ;
b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^A000085(j).

A293114 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 2, 6, 15, 45, 136, 430, 1415, 4845, 17235, 63509, 242854, 959904, 3926209, 16564083, 72097127, 322898943, 1487602607, 7034420691, 34122991199, 169499127425, 861596397518, 4475340840980, 23738200183570, 128427236055296, 708248486616539, 3977551340260517
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2017

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 2: {aa}, {ab}.
a(3) = 6: {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.

Links

Crossrefs

Main diagonal of A293112.
Row sums of A293113 and of A293815.
Cf. A000085, A182172, A293110, A306009.

Programs

  • Maple
    g:= proc(n) option remember;
      `if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
  • Mathematica
    g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* Binomial[g[i], j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

Formula

G.f.: Product_{j>=1} (1+x^j)^A000085(j).
Weigh transform of A000085.

A291057 Cardinality of the smallest nonempty class of length minimal languages with exactly n nonempty words each over a countably infinite alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 4, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, 5311735, 7726160, 9657700, 10400600, 9657700, 7726160, 5311735, 3124550, 1562275, 657800, 230230, 65780, 14950, 2600, 325, 26, 1
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Comments

a(n) is the smallest nonzero term in column n of A293815.

Examples

			a(0) = 1: {{}}.
a(1) = 1: {{a}}.
a(2) = 2: {{a,aa}, {a,ab}}.
a(3) = 1: {{a,aa,ab}}.
a(4) = 4: {{a,aa,ab,aaa}, {a,aa,ab,aab}, {a,aa,ab,aba}, {a,aa,ab,abc}}.
a(5) = 6: {{a,aa,ab,aaa,aab}, {a,aa,ab,aaa,aba}, {a,aa,ab,aaa,abc}, {a,aa,aab,aba}, {a,aa,ab,aab,ab,abc}, {a,aa,ab,aba,abc}}.
a(6) = 4: {{a,aa,ab,aaa,aab,aba}, {a,aa,ab,aaa,aab,abc}, {a,aa,ab,aaa,aba,abc}, {a,aa,ab,aab,aba,abc}}.
a(7) = 1: {{a,aa,ab,aaa,aab,aba,abc}}.
Breaking the sequence into lines after each 1 gives an irregular triangle whose j-th row equals the A000085(j)-th row of A007318 without its leftmost term. The leftmost column of this triangle is A000085:
   1;
   1;
   2,   1;
   4,   6,    4,     1;
  10,  45,  120,   210,   252,    210,    120,      45,      10,   1;
  26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, ...
  ...

Links

Crossrefs

Cf. A000085, A007318, A293815, A294129.

A294129 Numbers n for which exactly one length minimal language exists having exactly n nonempty words over a countably infinite alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

0, 1, 3, 7, 17, 43, 119, 351, 1115, 3735, 13231, 48927, 189079, 757583, 3148063, 13497599, 59704335, 271503647, 1268817471, 6078518911, 29837183007, 149789875903, 768674514815, 4026518397439, 21518708975039, 117199152735615, 650184360936191, 3670861106911743
Offset: 1

Views

Author

Alois P. Heinz, Oct 23 2017

Keywords

Comments

Numbers n such that A291057(n) equals 1.
Numbers n such that the smallest nonzero term in column n of A293815 equals 1.

Examples

			0 is a term because there is only one length minimal language with 0 words: {}.
1 is a term: {a}.
3 is a term: {a, aa, ab}.
7 is a term: {a, aa, ab, aaa, aab, aba, abc}.
17 is a term: {a, aa, ab, aaa, aab, aba, abc, aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abca, abcd}.

Links

Crossrefs

Cf. A000085, A245176, A291057, A293815.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, n*(n-1)/2,
      2*a(n-1)+(n-3)*a(n-2)-(n-2)*a(n-3))
    end:
seq(a(n), n=1..30);

Formula

a(n) = a(n-1) + A000085(n-1) for n>1, a(1) = 0.
a(n) = 2*a(n-1)+(n-3)*a(n-2)-(n-2)*a(n-3) for n>= 4, a(n) = n*(n-1)/2 for n<4.
a(n) = A245176(n-1) - 1 for n>0.

A293964 Number of sets of exactly two nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

2, 5, 18, 52, 168, 533, 1792, 6161, 22088, 81690, 313224, 1239532, 5068320, 21355130, 92714368, 413915690, 1899260064, 8941932948, 43168351136, 213385326440, 1079240048256, 5578228370404, 29443746273792, 158547032884868, 870370433845888, 4866859874106392
Offset: 3

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=2 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=3..30);

Formula

a(n) = [x^n y^2] Product_{j>=1} (1+y*x^j)^A000085(j).

A293965 Number of sets of exactly three nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 8, 30, 114, 411, 1462, 5237, 18998, 70220, 265010, 1024692, 4059100, 16504058, 68843340, 294854550, 1295771712, 5843980456, 27026394156, 128135282356, 622230803212, 3093321051636, 15728089431744, 81739630155456, 433801710925696, 2349410730317456
Offset: 5

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=3 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=5..30);

Formula

a(n) = [x^n y^3] Product_{j>=1} (1+y*x^j)^A000085(j).

A293966 Number of sets of exactly four nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

4, 22, 116, 482, 1966, 7682, 29845, 115438, 449870, 1770788, 7073065, 28727652, 118907910, 502249944, 2167410896, 9563204836, 43166853057, 199391604156, 942578850020, 4559743209208, 22566589645408, 114215149597312, 590875202641724, 3122678708581528
Offset: 8

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=4 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 5)
    end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=8..33);

Formula

a(n) = [x^n y^4] Product_{j>=1} (1+y*x^j)^A000085(j).

A293967 Number of sets of exactly five nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

6, 48, 274, 1338, 6035, 25874, 108002, 444458, 1818905, 7451418, 30693022, 127604480, 536876960, 2291507552, 9939572897, 43885543586, 197465168488, 906430558822, 4247727231198, 20333276583188, 99450038211268, 497066503157976, 2538584563166367
Offset: 11

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=5 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 6)
    end:
a:= n-> coeff(b(n$2), x, 5):
seq(a(n), n=11..35);

Formula

a(n) = [x^n y^5] Product_{j>=1} (1+y*x^j)^A000085(j).

A293968 Number of sets of exactly six nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

4, 62, 417, 2414, 12190, 57686, 260349, 1143710, 4936266, 21117128, 90035798, 384416432, 1649398948, 7133455202, 31173583589, 137947781614, 619247938106, 2824375268432, 13105785174035, 61940904739132, 298438345898409, 1466892183248186, 7358885205363735
Offset: 14

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=6 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=14..40);

Formula

a(n) = [x^n y^6] Product_{j>=1} (1+y*x^j)^A000085(j).

A293969 Number of sets of exactly seven nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 40, 394, 2766, 16251, 86162, 426894, 2021990, 9290152, 41829426, 185965908, 820999576, 3615595261, 15941247432, 70583512572, 314664832674, 1415621796873, 6439720543682, 29674662921377, 138736843637738, 659019083032289, 3184439719295586, 15669157686000028
Offset: 17

Views

Author

Alois P. Heinz, Oct 20 2017

Keywords

Links

Crossrefs

Column k=7 of A293815.
Cf. A000085.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 8)
    end:
a:= n-> coeff(b(n$2), x, 7):
seq(a(n), n=17..42);

Formula

a(n) = [x^n y^7] Product_{j>=1} (1+y*x^j)^A000085(j).
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