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A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A255970 #21 Sep 25 2023 17:33:26
%S A255970 1,0,1,0,2,2,0,3,8,6,0,5,24,42,24,0,7,60,198,264,120,0,11,144,780,
%T A255970 1848,1920,720,0,15,320,2778,10512,18840,15840,5040,0,22,702,9342,
%U A255970 53184,146760,208080,146160,40320,0,30,1486,30186,250128,999720,2129040,2479680,1491840,362880
%N A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A255970 Alois P. Heinz, <a href="/A255970/b255970.txt">Rows n = 0..140, flattened</a>
%F A255970 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i).
%F A255970 T(n,k) = k! * A256130(n,k).
%e A255970 T(3,1) = 3: 1a1a1a, 2a1a, 1a.
%e A255970 T(3,2) = 8: 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a.
%e A255970 T(3,3) = 6: 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
%e A255970 Triangle T(n,k) begins:
%e A255970   1;
%e A255970   0,  1;
%e A255970   0,  2,   2;
%e A255970   0,  3,   8,    6;
%e A255970   0,  5,  24,   42,    24;
%e A255970   0,  7,  60,  198,   264,    120;
%e A255970   0, 11, 144,  780,  1848,   1920,    720;
%e A255970   0, 15, 320, 2778, 10512,  18840,  15840,   5040;
%e A255970   0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320;
%e A255970   ...
%p A255970 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A255970       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A255970     end:
%p A255970 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A255970 seq(seq(T(n, k), k=0..n), n=0..10);
%t A255970 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k -i]*(-1)^i* Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A255970 Columns k=0-1 give: A000007, A000041 (for n>0).
%Y A255970 Main diagonal gives A000142.
%Y A255970 Row sums give A278644.
%Y A255970 Cf. A246935, A256130, A319600.
%K A255970 nonn,tabl
%O A255970 0,5
%A A255970 _Alois P. Heinz_, Mar 12 2015