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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.

%I A256116 #18 Oct 26 2018 09:53:59
%S A256116 1,0,1,0,1,2,0,1,9,10,0,1,34,112,84,0,1,125,930,1800,1008,0,1,461,
%T A256116 7018,26400,35640,15840,0,1,1715,51142,334152,816816,840840,308880,0,
%U A256116 1,6434,368464,3944220,15550080,27824160,23063040,7207200
%N 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.
%H A256116 Alois P. Heinz, <a href="/A256116/b256116.txt">Rows n = 0..140, flattened</a>
%F A256116 T(n,k) = (Sum_{i=0..k} (-1)^i * C(k,i) * A183135(n,k-i)) / A028310(k).
%F A256116 T(n,k) = (k-1)! * A256117(n,k) for k > 0.
%e A256116 T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
%e A256116 T(3,3) = 10: aabbcc, aabccb, aacbbc, aaccbb, abbacc, abbcca, abccba, acbbca, accabb, accbba.
%e A256116 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.
%e A256116 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.
%e A256116 Triangle T(n,k) begins:
%e A256116   1;
%e A256116   0, 1;
%e A256116   0, 1,    2;
%e A256116   0, 1,    9,    10;
%e A256116   0, 1,   34,   112,     84;
%e A256116   0, 1,  125,   930,   1800,   1008;
%e A256116   0, 1,  461,  7018,  26400,  35640,  15840;
%e A256116   0, 1, 1715, 51142, 334152, 816816, 840840, 308880;
%p A256116 A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
%p A256116       add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
%p A256116     end:
%p A256116 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/
%p A256116     `if`(k=0, 1, k):
%p A256116 seq(seq(T(n, k), k=0..n), n=0..12);
%t A256116 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 A256116 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 *)
%Y A256116 Columns k=0-2 give: A000007, A057427, A010763(n-1) for n>0.
%Y A256116 Main diagonal gives A065866(n-1) (for n>0).
%Y A256116 Row sums give A294603.
%Y A256116 Cf. A183135, A256117.
%K A256116 nonn,tabl
%O A256116 0,6
%A A256116 _Alois P. Heinz_, Mar 15 2015