A258461 -id:A258461 - cp's OEIS frontend

A256130 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.

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, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1

Offset: 0

Alois P. Heinz, Mar 15 2015

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

			T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
  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, 15,  160,   463,   438,   157,    22,    1;
  0, 22,  351,  1557,  2216,  1223,   289,   29,   1;
  0, 30,  743,  5031, 10422,  8331,  2957,  492,  37,  1;
  0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.
Row sums give A258466.
T(2n,n) give A258467.
Cf. A000142, A246935, A255970, A262495, A319730.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

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 *)

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

A320548 Number of partitions of n into parts of exactly six sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 16, 157, 1223, 8331, 52078, 307123, 1738442, 9552826, 51357799, 271624228, 1418856967, 7341442171, 37708533544, 192586163135, 979219603193, 4961598120154, 25071026570558, 126410385741982, 636282269651863, 3198360710675384, 16059685006324807

Offset: 6

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A262495.

Cf. A258461.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(6):
seq(a(n), n=6..40);

a(n) ~ 5^(n-1) / (5! * QPochhammer[1/5]). - Vaclav Kotesovec, Oct 25 2018

cp's OEIS Frontend

A256130 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.

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