|
%I #26 Jul 10 2018 19:16:01
%S 1,0,1,0,2,1,0,3,3,1,0,5,10,4,1,0,7,24,17,5,1,0,11,62,58,26,6,1,0,15,
%T 140,193,107,37,7,1,0,22,329,603,439,178,50,8,1,0,30,725,1852,1663,
%U 852,275,65,9,1,0,42,1631,5539,6283,3767,1500,402,82,10,1
%N Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A293109/b293109.txt">Rows n = 0..40, flattened</a>
%F T(n,k) = A293108(n,k) - A293108(n,k-1) for k>0, T(n,0) = A293108(n,0).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 3, 3, 1;
%e 0, 5, 10, 4, 1;
%e 0, 7, 24, 17, 5, 1;
%e 0, 11, 62, 58, 26, 6, 1;
%e 0, 15, 140, 193, 107, 37, 7, 1;
%e 0, 22, 329, 603, 439, 178, 50, 8, 1;
%p h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
%p <j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
%p g:= proc(n, i, l) option remember;
%p `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
%p g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p end:
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])
%p *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
%p end:
%p T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];
%t g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
%t A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
%t T[n_, 0] := A[n, 0]; T[n_, k_] := A[n, k] - A[n, k - 1];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 09 2018, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A000041 (for n>0), A293797, A293798, A293799, A293800, A293801, A293802, A293803, A293804, A293805.
%Y Row sums give A293110.
%Y T(2n,n) gives A293111.
%Y Cf. A182172, A293108, A293113.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Sep 30 2017
|
