A293108 -id:A293108 - cp's OEIS frontend

A293109 Number T(n,k) of multisets 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, 2, 1, 0, 3, 3, 1, 0, 5, 10, 4, 1, 0, 7, 24, 17, 5, 1, 0, 11, 62, 58, 26, 6, 1, 0, 15, 140, 193, 107, 37, 7, 1, 0, 22, 329, 603, 439, 178, 50, 8, 1, 0, 30, 725, 1852, 1663, 852, 275, 65, 9, 1, 0, 42, 1631, 5539, 6283, 3767, 1500, 402, 82, 10, 1

Offset: 0

Alois P. Heinz, Sep 30 2017

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   1;
  0,  3,   3,   1;
  0,  5,  10,   4,   1;
  0,  7,  24,  17,   5,   1;
  0, 11,  62,  58,  26,   6,  1;
  0, 15, 140, 193, 107,  37,  7, 1;
  0, 22, 329, 603, 439, 178, 50, 8, 1;

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

Columns k=0-10 give: A000007, A000041 (for n>0), A293797, A293798, A293799, A293800, A293801, A293802, A293803, A293804, A293805.
Row sums give A293110.
T(2n,n) gives A293111.
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:
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])
      *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
T[n_, 0] := A[n, 0]; T[n_, k_] := A[n, k] - A[n, k - 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2018, after Alois P. Heinz *)

T(n,k) = A293108(n,k) - A293108(n,k-1) for k>0, T(n,0) = A293108(n,0).

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).

A293732 Number of multisets of nonempty words with a total of n letters over binary 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, 6, 15, 31, 73, 155, 351, 755, 1673, 3604, 7897, 16988, 36902, 79222, 171030, 366180, 786746, 1679976, 3595207, 7657631, 16332935, 34706319, 73812099, 156503351, 332004423, 702533059, 1486998780, 3140716766, 6634315264, 13988517803, 29494816751

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3260 (terms 0..1000 from Alois P. Heinz)

Column k=2 of A293108.

Cf. A001405.

a:= proc(n) option remember; `if`(n=0, 1, add(add(binomial(d,
      floor(d/2))*d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

nmax = 40; A001405 = Table[Binomial[n, Floor[n/2]], {n, 1, nmax}]; CoefficientList[Series[Product[1/(1 - x^k)^A001405[[k]], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 30 2019 *)

G.f.: Product_{j>=1} 1/(1-x^j)^A001405(j).

a(n) ~ 2^(n - 1/6) * exp(3*(n/2)^(1/3) - 2 + S) / (sqrt(3*Pi) * n^(5/6)), where S = Sum_{k>=2} (sqrt(1/(1 - 1/2^(2*k - 2))) - 1) * (2^k + 2) / (2*k) = 0.3158684977247920135402311766405977266170498097655... - Vaclav Kotesovec, May 30 2019

A293735 Number of multisets of nonempty words with a total of n letters over quinary 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, 163, 492, 1571, 5122, 17262, 59483, 209958, 755615, 2770994, 10330036, 39103166, 150073289, 583329574, 2293822828, 9116935874, 36593731182, 148221246775, 605427601519, 2492286544749, 10334197803358, 43140208034891, 181224681022614

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

This sequence differs from A293110 first at n=6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A293108.

Cf. A049401, A293110, A293744.

g:= proc(n) option remember;
      `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*g(n-1)
       +(n-1)*(13*n+9)*g(n-2) -15*(n-1)*(n-2)*g(n-3)) /
       ((n+4)*(n+6)))
    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..35);

g[n_] := g[n] = If[n < 3, {1, 1, 2}[[n+1]], ((3n^2+17n+15) g[n-1] + (n-1)(13n+9) g[n-2] - 15(n-1)(n-2) g[n-3]) / ((n+4)(n+6))];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d] d, {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 35] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

G.f.: Product_{j>=1} 1/(1-x^j)^A049401(j).

a(n) ~ c * 5^n / n^5, where c = 542.824729617782144... - Vaclav Kotesovec, May 30 2019

A293736 Number of multisets of nonempty words with a total of n letters over senary 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, 499, 1621, 5397, 18762, 67000, 247439, 936167, 3639968, 14450634, 58677742, 242511781, 1021307520, 4365923278, 18960435664, 83395216882, 371734296357, 1675125941350, 7635063496721, 35127842511275, 163213032700613, 764541230737345

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

This sequence differs from A293110 first at n=7.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A293108.

Cf. A007579, A293110, A293745.

g:= proc(n) option remember;
      `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*g(n-1)
       +4*(n-1)*(10*n^2+58*n+33)*g(n-2) -144*(n-1)*(n-2)*g(n-3)
       -144*(n-1)*(n-2)*(n-3)*g(n-4))/ ((n+5)*(n+8)*(n+9)))
    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..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A007579(j).

a(n) ~ c * 6^n / n^(15/2), where c = 121210.8807171702661881473876689430182129891246619701141888082152779... - Vaclav Kotesovec, May 30 2019

A293737 Number of multisets of nonempty words with a total of n letters over septenary 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, 1629, 5462, 19164, 69457, 261154, 1012164, 4045640, 16611121, 70001515, 301922104, 1331128134, 5986321599, 27426419974, 127801386949, 605016657100, 2906093083727, 14149469612919, 69762426194708, 348016146152252, 1755188873640756

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

This sequence differs from A293110 first at n=8.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=7 of A293108.

Cf. A007578, A293110, A293746.

g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
      ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+
       305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
       *(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))
    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..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A007578(j).

a(n) ~ c * 7^n / n^(21/2), where c = 233774941.39802934196800791705821024006230754487492494942398064537776753785... - Vaclav Kotesovec, May 30 2019

A293738 Number of multisets of nonempty words with a total of n letters over octonary 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, 5471, 19246, 70020, 264961, 1035540, 4187725, 17440159, 74817905, 329400093, 1487844185, 6873585346, 32460719143, 156315314070, 767106102127, 3828629444020, 19423438144438, 99998608025751, 522200287437179, 2762351298913471

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

This sequence differs from A293110 first at n=9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A293108.

Cf. A007580, A293110, A293747.

g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
      ((40*n^3+1084*n^2+8684*n+18480)*g(n-1) +16*(n-1)*
      (5*n^3+107*n^2+610*n+600)*g(n-2) -1024*(n-1)*(n-2)*
      (n+6)*g(n-3) -1024*(n-1)*(n-2)*(n-3)*(n+4)*g(n-4))
       /((n+7)*(n+12)*(n+15)*(n+16)))
    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..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A007580(j).

a(n) ~ c * 8^n / n^14, where c = 4485962145436.6348123684794... - Vaclav Kotesovec, Dec 19 2020

A293739 Number of multisets of nonempty words with a total of n letters over nonary 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, 19256, 70121, 265723, 1041184, 4225484, 17689505, 76392933, 339283021, 1548592471, 7246210567, 34725853445, 170096822940, 850715642495, 4337263437693, 22519434992230, 118916581795167, 638088881489144, 3475474084479428

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

This sequence differs from A293110 first at n=10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=9 of A293108.

Cf. A212915, A293110, A293748.

G.f.: Product_{j>=1} 1/(1-x^j)^A212915(j).

a(n) ~ c * 9^n / n^18, where c = 978644175524508179.918397627321774... - Vaclav Kotesovec, Dec 19 2020

A293733 Number of multisets of nonempty words with a total of n letters over ternary 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, 19, 48, 131, 348, 954, 2607, 7212, 19995, 55816, 156246, 439267, 1238397, 3502004, 9927260, 28208628, 80322048, 229161413, 654966245, 1875074366, 5376298225, 15437286706, 44385247519, 127776425727, 368276055467, 1062618076382, 3069264747076

Offset: 0

Alois P. Heinz, Oct 15 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=3 of A293108.

Cf. A001006.

g:= proc(n) option remember; `if`(n<2, 1,
      g(n-1)+add(g(k)*g(n-k-2), k=0..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..35);

g[n_] := g[n] = If[n<2, 1, g[n-1] + Sum[g[k]*g[n-k-2], {k, 0, 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, 35}] (* Jean-François Alcover, May 30 2019, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import divisors
@cacheit
def g(n): return 1 if n<2 else g(n - 1) + sum([g(k)*g(n - k - 2) for k in range(n - 1)])
@cacheit
def a(n): return 1 if n==0 else sum([sum([g(d)*d for d in divisors(j)])*a(n - j) for j in range(1, n + 1)])//n
print(map(a, range(36))) # Indranil Ghosh, Oct 15 2017

G.f.: Product_{j>=1} 1/(1-x^j)^A001006(j).

a(n) ~ c * 3^n / n^(3/2), where c = 11.84175157992103588081767200532703865225243959779980786519467770732598276486... - Vaclav Kotesovec, May 30 2019

cp's OEIS Frontend

A293109 Number T(n,k) of multisets 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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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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A293732 Number of multisets of nonempty words with a total of n letters over binary 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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A293735 Number of multisets of nonempty words with a total of n letters over quinary 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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A293736 Number of multisets of nonempty words with a total of n letters over senary 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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A293737 Number of multisets of nonempty words with a total of n letters over septenary 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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A293738 Number of multisets of nonempty words with a total of n letters over octonary 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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