A293114 -id:A293114 - cp's OEIS frontend

A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter 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.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 2, 8, 4, 1, 0, 3, 20, 16, 5, 1, 0, 4, 47, 53, 25, 6, 1, 0, 5, 106, 173, 102, 36, 7, 1, 0, 6, 237, 532, 410, 172, 49, 8, 1, 0, 8, 522, 1615, 1545, 813, 268, 64, 9, 1, 0, 10, 1146, 4785, 5784, 3576, 1448, 394, 81, 10, 1

Offset: 0

Alois P. Heinz, Sep 30 2017

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 2,   3,   1;
  0, 2,   8,   4,   1;
  0, 3,  20,  16,   5,   1;
  0, 4,  47,  53,  25,   6,  1;
  0, 5, 106, 173, 102,  36,  7, 1;
  0, 6, 237, 532, 410, 172, 49, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..40, flattened

Columns k=0-10 give: A000007, A000009 (for n>0), A293883, A293884, A293885, A293886, A293887, A293888, A293889, A293890, A293891.
Row sums give A293114.
T(2n,n) gives A293115.
Cf. A182172, A293109, A293112.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
    end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

T(n,k) = A293112(n,k) - A293112(n,k-1) for k>0, T(n,0) = A293112(n,0).

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

Alois P. Heinz, Oct 16 2017

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

			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;
  ...

Alois P. Heinz, Rows n = 0..300, flattened

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.

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);

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 *)

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

A293112 Number A(n,k) of sets of nonempty words with a total of n letters over k-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; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 2, 0, 1, 1, 2, 6, 10, 3, 0, 1, 1, 2, 6, 14, 23, 4, 0, 1, 1, 2, 6, 15, 39, 51, 5, 0, 1, 1, 2, 6, 15, 44, 104, 111, 6, 0, 1, 1, 2, 6, 15, 45, 129, 284, 243, 8, 0, 1, 1, 2, 6, 15, 45, 135, 386, 775, 530, 10, 0

Offset: 0

Alois P. Heinz, Sep 30 2017

			Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2, ...
  0, 2,   5,   6,   6,   6,   6,   6, ...
  0, 2,  10,  14,  15,  15,  15,  15, ...
  0, 3,  23,  39,  44,  45,  45,  45, ...
  0, 4,  51, 104, 129, 135, 136, 136, ...
  0, 5, 111, 284, 386, 422, 429, 430, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0-10 give: A000007, A000009, A293741, A293742, A293743, A293744, A293745, A293746, A293747, A293748, A293749.
Main diagonal gives A293114.
Cf. A182172, A293108, A293113.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

G.f. of column k: Product_{j>=1} (1+x^j)^A182172(j,k).

A(n,k) = Sum_{j=0..k} A293113(n,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

Alois P. Heinz, Sep 30 2017

nonn

			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}.

Alois P. Heinz, Table of n, a(n) for n = 0..800

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

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);

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 *)

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

A306009 Inverse Weigh transform of A000085.

Original entry on oeis.org

1, 2, 2, 7, 14, 43, 130, 446, 1544, 5773, 22170, 89356, 370198, 1591379, 7020014, 31922981, 148679262, 710828036, 3474337098, 17379964444, 88739068866, 462670294023, 2458638559154, 13317850411827, 73432568553848, 412120738922369, 2351720323257872

Offset: 1

Alois P. Heinz, Jun 16 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..800

Cf. A000085, A293114.

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(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);

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[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Oct 27 2021, after Alois P. Heinz *)

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000085(n) * x^n.

cp's OEIS Frontend

A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter 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.

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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.

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A293112 Number A(n,k) of sets of nonempty words with a total of n letters over k-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; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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.

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A306009 Inverse Weigh transform of A000085.

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