A294603 -id:A294603

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A256116

Number T(n,k) of length 2n k-ary words, either empty or beginning with the first letter of the alphabet and using each letter at least once, that can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

+10
5

1, 0, 1, 0, 1, 2, 0, 1, 9, 10, 0, 1, 34, 112, 84, 0, 1, 125, 930, 1800, 1008, 0, 1, 461, 7018, 26400, 35640, 15840, 0, 1, 1715, 51142, 334152, 816816, 840840, 308880, 0, 1, 6434, 368464, 3944220, 15550080, 27824160, 23063040, 7207200

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

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

FORMULA

T(n,k) = (Sum_{i=0..k} (-1)^i * C(k,i) * A183135(n,k-i)) / A028310(k).

T(n,k) = (k-1)! * A256117(n,k) for k > 0.

EXAMPLE

T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.

T(3,3) = 10: aabbcc, aabccb, aacbbc, aaccbb, abbacc, abbcca, abccba, acbbca, accabb, accbba.

T(4,2) = 34: aaaaaabb, aaaaabba, aaaabaab, aaaabbaa, aaaabbbb, aaabaaba, aaabbaaa, aaabbabb, aaabbbba, aabaaaab, aabaabaa, aabaabbb, aababbab, aabbaaaa, aabbaabb, aabbabba, aabbbaab, aabbbbaa, aabbbbbb, abaaaaba, abaabaaa, abaababb, abaabbba, ababbaba, abbaaaaa, abbaaabb, abbaabba, abbabaab, abbabbaa, abbabbbb, abbbaaba, abbbbaaa, abbbbabb, abbbbbba.

T(4,4) = 84: aabbccdd, aabbcddc, aabbdccd, aabbddcc, aabccbdd, aabccddb, aabcddcb, aabdccdb, aabddbcc, aabddccb, aacbbcdd, aacbbddc, aacbddbc, aaccbbdd, aaccbddb, aaccdbbd, aaccddbb, aacdbbdc, aacddbbc, aacddcbb, aadbbccd, aadbbdcc, aadbccbd, aadcbbcd, aadccbbd, aadccdbb, aaddbbcc, aaddbccb, aaddcbbc, aaddccbb, abbaccdd, abbacddc, abbadccd, abbaddcc, abbccadd, abbccdda, abbcddca, abbdccda, abbddacc, abbddcca, abccbadd, abccbdda, abccddba, abcddcba, abdccdba, abddbacc, abddbcca, abddccba, acbbcadd, acbbcdda, acbbddca, acbddbca, accabbdd, accabddb, accadbbd, accaddbb, accbbadd, accbbdda, accbddba, accdbbda, accddabb, accddbba, acdbbdca, acddbbca, acddcabb, acddcbba, adbbccda, adbbdacc, adbbdcca, adbccbda, adcbbcda, adccbbda, adccdabb, adccdbba, addabbcc, addabccb, addacbbc, addaccbb, addbbacc, addbbcca, addbccba, addcbbca, addccabb, addccbba.

Triangle T(n,k) begins:

0, 1;

0, 1, 2;

0, 1, 9, 10;

0, 1, 34, 112, 84;

0, 1, 125, 930, 1800, 1008;

0, 1, 461, 7018, 26400, 35640, 15840;

0, 1, 1715, 51142, 334152, 816816, 840840, 308880;

MAPLE

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(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/

`if`(k=0, 1, k):

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

Unprotect[Power]; 0^0 = 1; A[n_, k_] := A[n, k] = If[n==0, 1, k/n*Sum[ Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]];

T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k==0, 1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

CROSSREFS

Columns k=0-2 give: A000007, A057427, A010763(n-1) for n>0.

Main diagonal gives A065866(n-1) (for n>0).

Row sums give A294603.

Cf. A183135, A256117.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 15 2015

STATUS

approved

A258498

Number of words of length 2n 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 doublets into the initially empty word.

+10
4

1, 1, 3, 15, 105, 933, 9988, 124449, 1761287, 27813479, 483482018, 9153385959, 187129080977, 4102129113670, 95861136747795, 2376234441556411, 62216635372018209, 1714347701138957189, 49553280367466054768, 1498300016807379304877, 47270249397381096576643

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..447

FORMULA

a(n) = Sum_{k=0..n} A256117(n,k).

a(n) ~ Bell(n-1)*Catalan(n) ~ n^n * exp(n/LambertW(n)-1-n) * 4^n / (sqrt(Pi) * sqrt(1+LambertW(n)) * LambertW(n)^(n-1) * n^(5/2)). - Vaclav Kotesovec, Jun 02 2015

EXAMPLE

a(3) = 15: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba, aabbcc, aabccb, abbacc, abbcca, abccba.

MAPLE

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):

a:= n-> add(T(n, k), k=0..n):

seq(a(n), n=0..25);

MATHEMATICA

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}];

a[n_] := Sum[T[n, k], {k, 0, n}];

a /@ Range[0, 25] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

CROSSREFS

Row sums of A256117.

Cf. A294603, A321031.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 31 2015

STATUS

approved

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