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.

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A292713 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 1055520, 10465854, 113479756, 1350508150, 17373376892, 241576630993, 3596468789967, 57232276979726, 967517444008250, 17339617861447844, 328037083000497867, 6537494747743375847, 136820214583596515519
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2017

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 4: {aa}, {ab}, {ba}, {a,a}.
a(3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.

Links

Crossrefs

Main diagonal of A292712.
Row sums of A319495.
Cf. A226873, A292796.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..25);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n];
a[n_] := A[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n] Product_{j=1..n} 1/(1-x^j)^A226873(j,n).
a(n) = A292712(n,n).
a(n) ~ c * n!, where c = A247551 = 2.5294774720791526... - Vaclav Kotesovec, Oct 05 2017

A226875 Number of n-length words w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 882, 3921, 18223, 84790, 432518, 1863951, 8892842, 42656147, 204204353, 1025014815, 4728033983, 22948258742, 111605089014, 541696830843, 2708218059022, 12861557284425, 62938669549583, 308273057334413, 1508708926286914, 7533652902408071
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Links

Crossrefs

Column k=5 of A226873.
Cf. A027306, A092255, A092429.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 5):
seq(a(n), n=0..30);
  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[If[i+j+k+l+m==n,n!/i!/j!/k!/l!/m!,0],{m,0,l}],{l,0,k}],{k,0,j}],{j,0,i}],{i,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 01 2013 *)
CoefficientList[Series[(HypergeometricPFQ[{},{},x]^5 + 10*HypergeometricPFQ[{},{},x]^3*HypergeometricPFQ[{},{1},x^2] + 20*HypergeometricPFQ[{},{},x]^2*HypergeometricPFQ[{},{1,1},x^3] + 20*HypergeometricPFQ[{},{1},x^2]*HypergeometricPFQ[{},{1,1},x^3] + 15*HypergeometricPFQ[{},{1},x^2]^2*HypergeometricPFQ[{},{},x] + 30*HypergeometricPFQ[{},{1,1,1},x^4]*HypergeometricPFQ[{},{},x] + 24*HypergeometricPFQ[{},{1,1,1,1},x^5])/5!,{x,0,20}],x]*Range[0,20]! (* more efficient, Vaclav Kotesovec, Jul 01 2013 *)

Formula

Conjecture: a(n) ~ 5^n/5!. - Vaclav Kotesovec, Mar 07 2014

A226876 Number of n-length words w over a 6-ary alphabet {a1,a2,...,a6} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a6) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 6441, 35023, 175510, 1017158, 5412111, 33991322, 168112907, 982269641, 5378704155, 31714236863, 174819971462, 1082436507990, 5756932808211, 34302363988462, 193719726696345, 1150224854410151, 6482217725030141, 39812123155826626
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Links

Crossrefs

Column k=6 of A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 6):
seq(a(n), n=0..30);

A226877 Number of n-length words w over a 7-ary alphabet {a1,a2,...,a7} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a7) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 55183, 326710, 1924358, 11843151, 76569242, 494393147, 3419744681, 20455085475, 133157018303, 860006815622, 5660947113750, 37583646117555, 249434965500622, 1713067949756985, 11030202759647591, 73747039462964885
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Links

Crossrefs

Column k=7 of A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 7):
seq(a(n), n=0..30);

A226878 Number of n-length words w over an 8-ary alphabet {a1,a2,...,a8} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a8) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 508150, 3436358, 21822351, 153741722, 1047906107, 7987668041, 57017211075, 456108767423, 3047668772102, 22857224364630, 163293406206195, 1236484989279502, 9040845014760345, 70057104400850471, 517521934394653205
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Links

Crossrefs

Column k=8 of A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 8):
seq(a(n), n=0..30);

A226879 Number of n-length words w over a 9-ary alphabet {a1,a2,...,a9} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a9) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 5250758, 38454351, 273492122, 2051148347, 15736849481, 125536061475, 1041102777023, 8537848507142, 74739775725270, 569218702884915, 4674633861692302, 37899687815748825, 312237339834676391, 2586068757754063445
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Links

Crossrefs

Column k=9 of A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 9):
seq(a(n), n=0..30);

A226880 Number of n-length words w over a 10-ary alphabet {a1,a2,...,a10} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a10) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 58412751, 473076122, 3607903547, 29782240841, 241773783075, 2137404383423, 18482746670342, 173563010955990, 1554987178737075, 15169020662626702, 126731980207937625, 1160565179374262951
Offset: 0

Views

Author

Alois P. Heinz, Jun 21 2013

Keywords

Comments

Differs from A005651 first at n=11: a(11) = 58412751 != A005651(11) = 98329551.

Links

Crossrefs

Column k=10 of A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 10):
seq(a(n), n=0..30);

A292796 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 3, 13, 60, 326, 2065, 14508, 116845, 1039459, 10339365, 112376487, 1339665295, 17256611005, 240193792120, 3578746993871, 56986570945387, 963868021665359, 17281651020455445, 327058650473873893, 6519981694119182165, 136489249161324882063
Offset: 0

Views

Author

Alois P. Heinz, Sep 23 2017

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 3: {aa}, {ab}, {ba}.
a(3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.

Links

Crossrefs

Main diagonal of A292795.
Row sums of A319498.
Cf. A226873, A292713.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
    end:
a:= n-> h(n$3):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
a[n_] := h[n, n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

a(n) = [x^n] Product_{j=1..n} (1+x^j)^A226873(j,n).
a(n) = A292795(n,n).
a(n) ~ c * n!, where c = A247551 = 2.529477472079152648... - Vaclav Kotesovec, Sep 28 2017

A292718 Number of multisets of nonempty words with a total of n letters over ternary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 43, 139, 495, 1544, 5111, 17348, 55520, 181946, 607300, 1951262, 6362769, 20972812, 67451405, 218884282, 715353298, 2298626230, 7429125757, 24124615697, 77400570114, 249285637563, 805472940377, 2579640351769, 8283108375403, 26655874638762
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2017

Keywords

Links

Crossrefs

Column k=3 of A292712.
Cf. A092255, A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 3), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A092255(j).
Euler transform of A092255.

A292719 Number of multisets of nonempty words with a total of n letters over quaternary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 223, 951, 3680, 16239, 61656, 260490, 1035820, 4451494, 17534372, 73518595, 295928531, 1253898892, 5015867442, 20920480946, 84742519783, 355861723649, 1434993799839, 5962065435072, 24234396539097, 101149561260620, 409761023233915
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2017

Keywords

Links

Crossrefs

Column k=4 of A292712.
Cf. A092429, A226873.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 4), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A092429(j).
Euler transform of A092429.
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