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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A319501 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the set; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 2, 12, 13, 0, 2, 38, 105, 73, 0, 3, 110, 588, 976, 501, 0, 4, 302, 2811, 8416, 9945, 4051, 0, 5, 806, 12354, 59488, 121710, 111396, 37633, 0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353, 0, 8, 5450, 207846, 2209276, 10096795, 23420022, 28969248, 18235680, 4596553
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2018

Keywords

Examples

			T(2,2) = 3: {ab}, {ba}, {a,b}.
T(3,2) = 12: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}.
T(4,2) = 38: {aaab}, {aaba}, {aabb}, {abaa}, {abab}, {abba}, {abbb}, {baaa}, {baab}, {baba}, {babb}, {bbaa}, {bbab}, {bbba}, {a,aab}, {a,aba}, {a,abb}, {a,baa}, {a,bab}, {a,bba}, {a,bbb}, {aa,ab}, {aa,ba}, {aa,bb}, {aaa,b}, {aab,b}, {ab,ba}, {ab,bb}, {aba,b}, {abb,b}, {b,baa}, {b,bab}, {b,bba}, {ba,bb}, {a,aa,b}, {a,ab,b}, {a,b,ba}, {a,b,bb}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    3;
  0, 2,   12,    13;
  0, 2,   38,   105,     73;
  0, 3,  110,   588,    976,     501;
  0, 4,  302,  2811,   8416,    9945,    4051;
  0, 5,  806, 12354,  59488,  121710,  111396,   37633;
  0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353;

Links

Crossrefs

Columns k=0-10 give: A000007, A000009 (for n>0), A320203, A320204, A320205, A320206, A320207, A320208, A320209, A320210, A320211.
Main diagonal gives A000262.
Row sums give A319518.
T(2n,n) gives A319519.
Cf. A257740, A292804.

Programs

  • Maple
    h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
T:= (n, k)-> add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    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[k^i, j], {j, 0, n/i}]]];
T[n_, k_] := Sum[(-1)^i Binomial[k, i] h[n, n, k-i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 05 2020, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A292804(n,k-i).

A257741 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that if a letter occurs in the multiset all predecessors occur at least once.

Original entry on oeis.org

1, 1, 5, 30, 241, 2356, 27315, 364319, 5488468, 92040141, 1698933390, 34206221161, 745622368096, 17486274798203, 438859174516837, 11732964019785027, 332818604033186036, 9981540739647177238, 315518234680527952625, 10482878954868309043158, 365158449014981632341391
Offset: 0

Views

Author

Alois P. Heinz, May 06 2015

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 5: {a,a}, {aa}, {ab}, {ba}, {a,b}.

Links

Crossrefs

Cf. A257740, A319518.

Programs

  • Mathematica
    A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^#&]*A[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A257740 *)

Formula

a(n) = Sum_{k=0..n} A257740(n,k).

A320265 Number of proper multisets of nonempty words with a total of n letters over n-ary alphabet such that if a letter occurs in the multiset all predecessors occur at least once.

Original entry on oeis.org

1, 3, 23, 178, 1786, 20927, 282520, 4299263, 72750927, 1353700567, 27452623890, 602326265519, 14209892886819, 358576428141962, 9634718410829852, 274567642777650028, 8270000441627265937, 262464788618069324640, 8752908129221863491691, 305968679117675345995513
Offset: 2

Views

Author

Alois P. Heinz, Oct 08 2018

Keywords

Examples

			a(2) = 1: {a,a}.
a(3) = 3: {a,a,a}, {a,a,b}, {a,b,b}.
a(4) = 23: {a,a,a,a}, {a,a,aa}, {aa, aa}, {a,a,a,b}, {a,a,b,b}, {a,b,b,b}, {a,a,ab}, {a,a,ba}, {a,a,bb}, {b,b,ab}, {b,b,ba}, {b,b,aa}, {ab,ab}, {ba,ba}, {a,a,b,c}, {a,a,bc}, {a,a,cb}, {b,b,a,c}, {b,b,ac}, {b,b,ca}, {c,c,a,b}, {c,c,ab}, {c,c,ba}.

Links

Crossrefs

Row sums of A320264.
Cf. A257741, A319518.

Programs

  • Maple
    h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> add(add((-1)^i*(g(n, k-i)-h(n$2, k-i))*
        binomial(k, i), i=0..k), k=1..n-1):
seq(a(n), n=2..25);
  • Mathematica
    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[k^i, j], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 1, n - 1}];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A320264 *)

Formula

a(n) = Sum_{k=1..n-1} A320264(n,k).
a(n) = A257741(n) - A319518(n).
Showing 1-3 of 3 results.

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