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A257742 Number of multisets of nonempty words with a total of 2n letters over n-ary alphabet such that all n letters occur at least once in the multiset.

%I A257742 #16 May 10 2022 14:51:33
%S A257742 1,2,49,3334,428653,87804401,26047147641,10515038040403,
%T A257742 5527943088161719,3662449762145471938,2981185419002290273673,
%U A257742 2921408464370908053081409,3389743512704136305019696050,4593040689601644978081159072298,7182956101782940369861692674495595
%N A257742 Number of multisets of nonempty words with a total of 2n letters over n-ary alphabet such that all n letters occur at least once in the multiset.
%H A257742 Alois P. Heinz, <a href="/A257742/b257742.txt">Table of n, a(n) for n = 0..200</a>
%F A257742 a(n) = A257740(2n,n).
%e A257742 a(0) = 1: {}.
%e A257742 a(1) = 2: {aa}, {a,a}.
%e A257742 a(2) = 49: {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,ab}, {ab,ba}, {ab,bb}, {aba,b}, {abb,b}, {b,baa}, {b,bab}, {b,bba}, {ba,ba}, {ba,bb}, {a,a,ab}, {a,a,ba}, {a,a,bb}, {a,aa,b}, {a,ab,b}, {a,b,ba}, {a,b,bb}, {aa,b,b}, {ab,b,b}, {b,b,ba}, {a,a,a,b}, {a,a,b,b}, {a,b,b,b}.
%t A257742 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^# &]*
%t A257742      A[n - j, k], {j, 1, n}]/n];
%t A257742 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A257742 a[n_] := T[2n, n]; Table[a[n], {n, 0, 15}]
%t A257742 (* _Jean-François Alcover_, May 10 2022, after _Alois P. Heinz_ in A257740 *)
%Y A257742 Cf. A257740, A319519.
%K A257742 nonn
%O A257742 0,2
%A A257742 _Alois P. Heinz_, May 06 2015
%E A257742 New name from _Alois P. Heinz_, Sep 21 2018