A292804 -id:A292804 - cp's OEIS frontend

A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

1, 2, 5, 16, 42, 116, 310, 816, 2121, 5466, 13937, 35248, 88494, 220644, 546778, 1347344, 3302780, 8057344, 19568892, 47329264, 114025786, 273709732, 654765342, 1561257968, 3711373005, 8797021714, 20794198581, 49024480880, 115292809910, 270495295636

Offset: 0

Philippe Flajolet, Mar 01 2005

nonn

Analogous to A034899 (which also enumerates multisets of words)

			a(2) = 5 because the sets are {a,b}, {aa}, {ab}, {ba}, {bb}.
a(3) = 16 because the sets are {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 64
Stefan Gerhold, Counting finite languages by total word length, INTEGERS 11 (2011), #A44.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A000079, A034899, A256142.
Column k=2 of A292804.
Row sums of A208741 and of A360634.

series(exp(add((-1)^(j-1)/j*(2*z^j)/(1-2*z^j),j=1..40)),z,40);

nn = 20; p = Product[(1 + x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
CoefficientList[Series[E^Sum[(-1)^(k-1)/k*(2*x^k)/(1-2*x^k), {k,1,30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 13 2014 *)

G.f.: exp(Sum((-1)^(j-1)/j*(2*z^j)/(1-2*z^j), j=1..infinity)).
Asymptotics (Gerhold, 2011): a(n) ~ c * 2^(n-1)*exp(2*sqrt(n)-1/2) / (sqrt(Pi) * n^(3/4)), where c = exp( Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1)) ) = 0.6602994483152065685... . - Vaclav Kotesovec, Sep 13 2014
Weigh transform of A000079. - Alois P. Heinz, Jun 25 2018

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

Alois P. Heinz, Sep 20 2018

			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;

Alois P. Heinz, Rows n = 0..140, flattened

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.

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);

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 *)

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

A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126

Offset: 0

Alois P. Heinz, Sep 09 2008

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

			A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
  1, 1,   1,    1,    1,     1, ...
  0, 1,   2,    3,    4,     5, ...
  0, 2,   7,   15,   26,    40, ...
  0, 3,  20,   64,  148,   285, ...
  0, 5,  59,  276,  843,  2020, ...
  0, 7, 162, 1137, 4632, 13876, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Cf. A004248, A257740, A256142, A292804.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, ] = 1; a[?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

Name changed by Alois P. Heinz, Sep 21 2018

A292795 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each 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, 3, 2, 0, 1, 1, 3, 7, 2, 0, 1, 1, 3, 13, 18, 3, 0, 1, 1, 3, 13, 36, 42, 4, 0, 1, 1, 3, 13, 60, 122, 110, 5, 0, 1, 1, 3, 13, 60, 206, 433, 250, 6, 0, 1, 1, 3, 13, 60, 326, 865, 1356, 627, 8, 0, 1, 1, 3, 13, 60, 326, 1345, 3408, 4449, 1439, 10, 0

Offset: 0

Alois P. Heinz, Sep 23 2017

			A(2,3) = 3: {aa}, {ab}, {ba}.
A(3,2) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
A(3,3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.
Square array A(n,k) begins:
  1, 1,   1,    1,     1,     1,     1,     1,      1, ...
  0, 1,   1,    1,     1,     1,     1,     1,      1, ...
  0, 1,   3,    3,     3,     3,     3,     3,      3, ...
  0, 2,   7,   13,    13,    13,    13,    13,     13, ...
  0, 2,  18,   36,    60,    60,    60,    60,     60, ...
  0, 3,  42,  122,   206,   326,   326,   326,    326, ...
  0, 4, 110,  433,   865,  1345,  2065,  2065,   2065, ...
  0, 5, 250, 1356,  3408,  6228,  9468, 14508,  14508, ...
  0, 6, 627, 4449, 15025, 29845, 51325, 76525, 116845, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000009, A340409, A340410, A340411, A340412, A340413, A340414, A340415, A340416, A340417.
Rows n=0-1 give: A000012, A057427.
Main diagonal gives A292796.
Cf. A226873, A292712, A292804, A319498.

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, k)-> h(n$2, min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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_, k_] := h[n, n, Min[n, k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten(* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} (1+x^j)^A226873(j,k).

A(n,k) = Sum_{j=0..n} A319498(n,j).

A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

Original entry on oeis.org

1, 3, 12, 55, 225, 927, 3729, 14787, 57888, 224220, 860022, 3270744, 12343899, 46264257, 172305837, 638039136, 2350109736, 8613851832, 31428857611, 114187160631, 413222547846, 1489829356657, 5352683946903, 19167988920930, 68427472477338, 243559693397025

Offset: 0

Vaclav Kotesovec, Mar 16 2015

nonn

In general, if g.f. = Product_{j>=1} (1+x^j)^(k^j), then a(n) ~ k^n * exp(2*sqrt(n) - 1/2 - c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} (-1)^m/(m*(k^(m-1)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of sequence A034691
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A102866, A144067, A144074.

Column k=3 of A292804.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .

A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.

Original entry on oeis.org

1, 1, 5, 55, 729, 12376, 250735, 5904746, 158210353, 4747112731, 157545928646, 5726207734545, 226093266070501, 9632339536696943, 440262935648935344, 21482974431740480311, 1114363790702406540897, 61219233429920494716931, 3550130647865299090804375

Offset: 0

Alois P. Heinz, Sep 23 2017

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..381

Main diagonal of A292804.

Cf. A252654, A292845.

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:
a:= n-> h(n$3):
seq(a(n), n=0..20);

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}]]];
a[n_] := h[n, n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

a(n) = [x^n] Product_{j=1..n} (1+x^j)^(n^j).

a(n) ~ n^(n - 3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A292845 Total number of words beginning with the first letter of an n-ary alphabet in all sets of nonempty words with a total of n letters.

Original entry on oeis.org

0, 1, 3, 28, 325, 4976, 92869, 2038842, 51397801, 1461081781, 46192638386, 1606531631321, 60921659773609, 2500525907856718, 110403919405245712, 5216038547426332891, 262495788417549517393, 14015335940464667636300, 791161963786588958170705

Offset: 0

Alois P. Heinz, Sep 24 2017

nonn

			For n=2 and alphabet {a,b} we have 5 sets: {aa}, {ab}, {ba}, {bb}, {a,b}. There is a total of 3 words beginning with the first alphabet letter, thus a(2) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..382

Cf. A216158, A292804, A292805, A292873.

h:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add(
     (p-> p+[0, p[1]*j])(binomial(k^i, j)*h(n-i*j, i-1, k)), j=0..n/i)))
    end:
a= n-> `if`(n=0, 0, h(n$3)[2]/n):
seq(a(n), n=0..22);

h[n_, i_, k_] := h[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[ Function[p, p + {0, p[[1]]*j}][Binomial[k^i, j]*h[n - i*j, i - 1, k]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 0, h[n, n, n][[2]]/n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

A292838 Number of sets of nonempty words with a total of n letters over quaternary alphabet.

Original entry on oeis.org

1, 4, 22, 132, 729, 4000, 21488, 113760, 594548, 3073392, 15732936, 79846448, 402104884, 2010879968, 9992425872, 49366096352, 242584319710, 1186177166680, 5773569726884, 27982357252632, 135079969593838, 649640609539360, 3113354757088720, 14871179093155424

Offset: 0

Alois P. Heinz, Sep 24 2017

nonn

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

Column k=4 of A292804.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(4^i, j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..30);

h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[h[n - i j,  i - 1] Binomial[4^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f.: Product_{j>=1} (1+x^j)^(4^j).

a(n) ~ 4^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(4^(m-1)-1)) = 0.147762663788961720137665013823002812172... - Vaclav Kotesovec, Sep 28 2017

A292839 Number of sets of nonempty words with a total of n letters over 5-ary alphabet.

Original entry on oeis.org

1, 5, 35, 260, 1805, 12376, 83175, 550775, 3600400, 23276175, 149012380, 945726575, 5955676150, 37243117575, 231412658225, 1429522303905, 8783382129825, 53700395135475, 326809026132350, 1980383108328950, 11952682268739660, 71870696616619250, 430632502970026125

Offset: 0

Alois P. Heinz, Sep 24 2017

nonn

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

Column k=5 of A292804.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(5^i, j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..30);

h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[h[n - i j, i - 1] Binomial[5^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f.: Product_{j>=1} (1+x^j)^(5^j).

a(n) ~ 5^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(5^(m-1)-1)) = 0.112852293193143374268678097722831649456... - Vaclav Kotesovec, Sep 28 2017

A292840 Number of sets of nonempty words with a total of n letters over 6-ary alphabet.

Original entry on oeis.org

1, 6, 51, 452, 3777, 31074, 250735, 1993176, 15640983, 121378650, 932738805, 7105552308, 53709133137, 403124780178, 3006420386499, 22290321581448, 164378277308862, 1206180958964508, 8810022165617086, 64073173243207632, 464122836576398454, 3349321050668452460

Offset: 0

Alois P. Heinz, Sep 24 2017

nonn

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

Column k=6 of A292804.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(6^i, j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..30);

G.f.: Product_{j>=1} (1+x^j)^(6^j).

a(n) ~ 6^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(6^(m-1)-1)) = 0.091503304254691843343610606469481430508... - Vaclav Kotesovec, Sep 28 2017

cp's OEIS Frontend

A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

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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.

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A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A292795 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each 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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Maple

Mathematica

Formula

A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

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Mathematica

Formula

A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.

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Maple

Mathematica

Formula

A292845 Total number of words beginning with the first letter of an n-ary alphabet in all sets of nonempty words with a total of n letters.

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Maple

Mathematica

A292838 Number of sets of nonempty words with a total of n letters over quaternary alphabet.