A256117
Number T(n,k) of length 2n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 2, 0, 1, 9, 5, 0, 1, 34, 56, 14, 0, 1, 125, 465, 300, 42, 0, 1, 461, 3509, 4400, 1485, 132, 0, 1, 1715, 25571, 55692, 34034, 7007, 429, 0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430, 0, 1, 24309, 1325609, 7488228, 11187462, 6191808, 1447992, 143208, 4862
Offset: 0
T(0,0) = 1: (the empty word).
T(1,1) = 1: aa.
T(2,1) = 1: aaaa.
T(2,2) = 2: aabb, abba.
T(3,1) = 1: aaaaaa.
T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
T(3,3) = 5: aabbcc, aabccb, abbacc, abbcca, abccba.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 9, 5;
0, 1, 34, 56, 14;
0, 1, 125, 465, 300, 42;
0, 1, 461, 3509, 4400, 1485, 132;
0, 1, 1715, 25571, 55692, 34034, 7007, 429;
0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430;
...
Columns k=0-10 give:
A000007,
A057427,
A010763(n-1) (for n>1),
A258490,
A258491,
A258492,
A258493,
A258494,
A258495,
A258496,
A258497.
-
A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
-
A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz, updated Jan 01 2021 *)
A294603
Number of words of semilength n over n-ary alphabet, either empty or beginning with the first letter of the alphabet, such that the index set of occurring letters is an integer interval [1, k], that can be built by repeatedly inserting doublets into the initially empty word.
Original entry on oeis.org
1, 1, 3, 20, 231, 3864, 85360, 2353546, 77963599, 3019479344, 133966276692, 6702399275538, 373406941221160, 22930441709648290, 1539004344848618466, 112089683771614695478, 8805334896381292460191, 742162775145283382779168, 66809386370870410069346476
Offset: 0
a(0) = 1: the empty word.
a(1) = 1: aa.
a(2) = 3: aaaa, aabb, abba.
a(3) = 20: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, aabbcc, aabccb, aacbbc, aaccbb, abaaba, abbaaa, abbabb, abbacc, abbbba, abbcca, abccba, acbbca, accabb, accbba.
-
A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
end:
T:= proc(n, k) option remember;
add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/`if`(k=0, 1, k)
end:
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..20);
-
A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*
Sum[Binomial[2*n, j]*(n-j) *If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := T[n, k] =
Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k == 0, 1, k];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)
A321031
Number of words of length 3n such that the index set of occurring letters is {1, 2, ..., k}, all letters are introduced in ascending order, and the words can be built by repeatedly inserting triples into the initially empty word.
Original entry on oeis.org
1, 1, 4, 31, 351, 5144, 91816, 1918578, 45687682, 1216354021, 35689352250, 1141323078031, 39429988969021, 1461049507764175, 57720478019188989, 2419008380691088543, 107083662651332423339, 4988596265684542112304, 243781041304397011647766
Offset: 0
a(2) = 4: aaaaaa, aaabbb, aabbba, abbbaa.
-
a:= n-> `if`(n=0, 1, add(add((-1)^i*(k-i)/n*add(binomial(3*n, j)
*(n-j)*(k-i-1)^j, j=0..n-1)/(i!*(k-i)!), i=0..k), k=0..n)):
seq(a(n), n=0..20);
Showing 1-3 of 3 results.
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