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A215521 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A215521 #30 Aug 26 2019 17:51:34
%S A215521 1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,3,2,1,1,1,3,4,3,2,1,1,1,4,5,5,3,2,
%T A215521 1,1,1,4,7,6,5,3,2,1,1,1,5,8,9,7,5,3,2,1,1,1,5,10,10,10,7,5,3,2,1,1,1,
%U A215521 6,12,14,12,11,7,5,3,2,1,1,1,6,14,16,17,13,11,7,5,3,2,1,1
%N A215521 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%C A215521 Differs from A008284 first at T(11,4).
%H A215521 Alois P. Heinz, <a href="/A215521/b215521.txt">Rows n = 1..100, flattened</a>
%H A215521 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%e A215521 T(4,2) = 2 = |{4!/(2!*2!), 4!/(2!*1!*1!)}| = |{6, 12}|.
%e A215521 T(7,4) = 3 = |{35, 105, 210}|.
%e A215521 T(8,3) = 5 = |{560, 1120, 1680, 3360, 6720}|.
%e A215521 T(11,4) = 10 = |{11550, 34650, 46200, 69300, 138600, 207900, 277200, 415800, 831600, 1663200}|.
%e A215521 Triangle T(n,k) begins:
%e A215521   1;
%e A215521   1,  1;
%e A215521   1,  1,  1;
%e A215521   1,  2,  1,  1;
%e A215521   1,  2,  2,  1,  1;
%e A215521   1,  3,  3,  2,  1,  1;
%e A215521   1,  3,  4,  3,  2,  1,  1;
%e A215521   1,  4,  5,  5,  3,  2,  1,  1;
%e A215521   1,  4,  7,  6,  5,  3,  2,  1,  1;
%e A215521   1,  5,  8,  9,  7,  5,  3,  2,  1,  1;
%e A215521   1,  5, 10, 10, 10,  7,  5,  3,  2,  1,  1;
%e A215521   ...
%p A215521 b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
%p A215521       {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
%p A215521     end:
%p A215521 T:= (n, k)-> nops(b(n-k, min(k, n-k))):
%p A215521 seq(seq(T(n, k), k=1..n), n=1..15);
%t A215521 b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n - k, Min[k, n - k]], {k, 1, n}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Jan 21 2015, after _Alois P. Heinz_ *)
%Y A215521 Columns k=1-3 give: A000012 (for n>0), A004526, A069905(n) = A001399(n-3) for n>=3.
%Y A215521 T(2*n,n) gives: A070289.
%Y A215521 Cf. A008284, A215520.
%K A215521 nonn,tabl
%O A215521 1,8
%A A215521 _Alois P. Heinz_, Aug 14 2012