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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.
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
OFFSET
0,6
LINKS
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:
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;
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.
Sequence in context: A352372 A219034 A372244 * A185410 A264676 A091803
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 15 2015
STATUS
approved