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 12 results. Next

A293815 Number T(n,k) of sets 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, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 2, 0, 10, 5, 0, 26, 18, 1, 0, 76, 52, 8, 0, 232, 168, 30, 0, 764, 533, 114, 4, 0, 2620, 1792, 411, 22, 0, 9496, 6161, 1462, 116, 0, 35696, 22088, 5237, 482, 6, 0, 140152, 81690, 18998, 1966, 48, 0, 568504, 313224, 70220, 7682, 274
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2017

Keywords

Comments

The smallest nonzero term in column k is A291057(k).

Examples

			T(0,0) = 1: {}.
T(3,1) = 4: {aaa}, {aab}, {aba}, {abc}.
T(3,2) = 2: {a,aa}, {a,ab}.
T(4,2) = 5: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,ab}.
T(5,3) = 1: {a,aa,ab}.
Triangle T(n,k) begins:
  1;
  0,      1;
  0,      2;
  0,      4,     2;
  0,     10,     5;
  0,     26,    18,     1;
  0,     76,    52,     8;
  0,    232,   168,    30;
  0,    764,   533,   114,    4;
  0,   2620,  1792,   411,   22;
  0,   9496,  6161,  1462,  116;
  0,  35696, 22088,  5237,  482,  6;
  0, 140152, 81690, 18998, 1966, 48;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000085 (for n>0), A293964, A293965, A293966, A293967, A293968, A293969, A293970, A293971, A293972.
Row sums give A293114.
Cf. A208741, A293808, A291057, A294129.

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, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(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, 1, If[i<1, 0, Sum[b[n - i*j, i-1]* Binomial[g[i], j]*x^j, {j, 0, n/i}]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ 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+y*x^j)^A000085(j).

A293110 Number of multisets 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, 3, 7, 20, 54, 164, 500, 1630, 5472, 19257, 70133, 265858, 1042346, 4235031, 17760943, 76913277, 342919431, 1573637985, 7415371293, 35860511131, 177641956111, 900782461170, 4668600610346, 24714284921937, 133467868645017, 734844788634269, 4120752558254581
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2017

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: {a}
a(2) = 3: {a,a}, {aa}, {ab}.
a(3) = 7: {a,a,a}, {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.

Links

Crossrefs

Main diagonal of A293108.
Row sums of A293109 and of A293808.
Cf. A000085, A182172, A293114.

Programs

  • Maple
    g:= proc(n) option remember;
      `if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(g(d)
      *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);
  • Mathematica
    g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d]*d, {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 07 2018, from Maple *)

Formula

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

A293809 Number of multisets of exactly n nonempty words with a total of 2n letters over 2n-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, 2, 7, 22, 73, 240, 818, 2824, 10004, 36252, 134594, 512632, 2002797, 8037634, 33122211, 140287074, 610344666, 2728599114, 12524559427, 59014996342, 285169596358, 1412357461074, 7161541766341, 37150562120334, 196945057245451, 1066104659977212
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2017

Keywords

Examples

			a(0) = 1: {}.
a(1) = 2: {aa}, {ab}.
a(2) = 7: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,aa}, {aa,ab}, {ab,ab}.

Links

Crossrefs

Cf. A000085, A293808.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      g(d+1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);
  • Mathematica
    g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2023, after Alois P. Heinz *)

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A000085(j+1).
Euler transform of j-> A000085(j+1).
a(n) = A293808(2n,n).

A294004 Number of multisets 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

1, 2, 7, 18, 56, 168, 543, 1792, 6187, 22088, 81766, 313224, 1239764, 5068320, 21355894, 92714368, 413918310, 1899260064, 8941942444, 43168351136, 213385362136, 1079240048256, 5578228510556, 29443746273792, 158547033453372, 870370433845888, 4866859876496872
Offset: 2

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=2 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..30);

Formula

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

A294005 Number of multisets 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, 2, 7, 22, 68, 218, 721, 2438, 8491, 30478, 112524, 428382, 1678600, 6778708, 28169286, 120516092, 530081370, 2396797920, 11125584584, 52993063796, 258676491628, 1293160049244, 6612750833996, 34564483264256, 184470133103464, 1004514566402816
Offset: 3

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=3 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..30);

Formula

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

A294006 Number of multisets 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

1, 2, 7, 22, 73, 234, 791, 2702, 9507, 34258, 126807, 482306, 1885031, 7578028, 31316391, 133117500, 581531653, 2611112712, 12037781812, 56962049532, 276345797775, 1373655295948, 6988160240848, 36356528106984, 193225799686632, 1048279646446240
Offset: 4

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=4 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 5)
    end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=4..35);

Formula

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

A294007 Number of multisets 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

1, 2, 7, 22, 73, 240, 811, 2792, 9857, 35644, 132119, 502832, 1964131, 7885792, 32523695, 137915764, 600865387, 2690302074, 12367812720, 58364059306, 282421855885, 1400551909446, 7109841300492, 36919536804334, 195890584265442, 1061185175436116
Offset: 5

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=5 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 6)
    end:
a:= n-> coeff(b(n$2), x, 5):
seq(a(n), n=5..35);

Formula

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

A294008 Number of multisets 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

1, 2, 7, 22, 73, 240, 818, 2816, 9967, 36080, 133875, 509676, 1990984, 7990628, 32936173, 139548808, 607402437, 2716780286, 12476624346, 58818236078, 284350933608, 1408898449946, 7146679566822, 37085526689402, 196654885016221, 1064783059174600
Offset: 6

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=6 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=6..35);

Formula

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

A294009 Number of multisets 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, 2, 7, 22, 73, 240, 818, 2824, 9995, 36210, 134397, 511802, 1999360, 8023808, 33066865, 140066840, 609466485, 2725084766, 12510393090, 58957378290, 284932585092, 1411369884766, 7157365741706, 37132616218394, 196866561660145, 1065754768886044
Offset: 7

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=7 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 8)
    end:
a:= n-> coeff(b(n$2), x, 7):
seq(a(n), n=7..40);

Formula

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

A294010 Number of multisets of exactly eight 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, 2, 7, 22, 73, 240, 818, 2824, 10004, 36242, 134547, 512410, 2001856, 8033716, 33106372, 140223388, 610090236, 2727581018, 12520472740, 58998480846, 285102284159, 1412080134386, 7160384929556, 37145667315382, 196924018956010, 1066012662681880
Offset: 8

Views

Author

Alois P. Heinz, Oct 21 2017

Keywords

Links

Crossrefs

Column k=8 of A293808.
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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 9)
    end:
a:= n-> coeff(b(n$2), x, 8):
seq(a(n), n=8..40);

Formula

a(n) = [x^n y^8] Product_{j>=1} 1/(1-y*x^j)^A000085(j).
Showing 1-10 of 12 results. Next

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