A256130 - OEIS

%I #28 Feb 14 2024 16:52:09

%S 1,0,1,0,2,1,0,3,4,1,0,5,12,7,1,0,7,30,33,11,1,0,11,72,130,77,16,1,0,

%T 15,160,463,438,157,22,1,0,22,351,1557,2216,1223,289,29,1,0,30,743,

%U 5031,10422,8331,2957,492,37,1,0,42,1561,15877,46731,52078,26073,6401,788,46,1

%N Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%C In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - _Vaclav Kotesovec_, Jun 01 2015

%H Alois P. Heinz, <a href="/A256130/b256130.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = A255970(n,k)/k! = (Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i)) / A000142(k).

%e T(3,1) = 3: 1a1a1a, 2a1a, 3a.

%e T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.

%e T(3,3) = 1: 1a1b1c.

%e Triangle T(n,k) begins:

%e   1;

%e   0,  1;

%e   0,  2,    1;

%e   0,  3,    4,     1;

%e   0,  5,   12,     7,     1;

%e   0,  7,   30,    33,    11,     1;

%e   0, 11,   72,   130,    77,    16,     1;

%e   0, 15,  160,   463,   438,   157,    22,    1;

%e   0, 22,  351,  1557,  2216,  1223,   289,   29,   1;

%e   0, 30,  743,  5031, 10422,  8331,  2957,  492,  37,  1;

%e   0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46,  1;

%e   ...

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))

%p     end:

%p T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 21 2016, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.

%Y Row sums give A258466.

%Y T(2n,n) give A258467.

%Y Cf. A000142, A246935, A255970, A262495, A319730.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Mar 15 2015