A320546 -id:A320546 - cp's OEIS frontend

A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 6, 12, 7, 1, 0, 1, 10, 31, 33, 11, 1, 0, 1, 14, 73, 130, 77, 16, 1, 0, 1, 21, 165, 464, 438, 157, 22, 1, 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1, 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1

Offset: 0

Alois P. Heinz, Sep 24 2015

			T(3,1) = 1: 3a.
T(3,2) = 2: 2a1b, 1a1b1a.
T(3,3) = 1: 1a1b1c.
T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  4,   4,    1;
  0, 1,  6,  12,    7,     1;
  0, 1, 10,  31,   33,    11,    1;
  0, 1, 14,  73,  130,    77,   16,    1;
  0, 1, 21, 165,  464,   438,  157,   22,   1;
  0, 1, 29, 357, 1558,  2216, 1223,  289,  29,  1;
  0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
  ...

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

Columns k=0-10 give: A000007, A057427, A000065, A262445/6 = A320545, A320546, A320547, A320548, A320549, A320550, A320551, A320552.
Main diagonal and lower diagonal give: A000012, A000124 (shifted).
Row sums give A262496.
T(2n,n) gives A262529.
Cf. A256130.

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))):
T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *)

A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 11, 77, 438, 2216, 10422, 46731, 202814, 860586, 3593561, 14834956, 60735095, 247155292, 1001318246, 4043482110, 16288762319, 65500024027, 263035832734, 1055252430510, 4230340216034, 16949359882259, 67881449170593, 271777855641517, 1087867649157513

Offset: 4

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A256130.

Cf. A320546.

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):
a:= n-> T(n,4):
seq(a(n), n=4..35);

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[T[n, 4], {n, 4, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

a(n) ~ c * 4^n, where c = 1/(24*Product_{n>=1} (1-1/4^n)) = 1/(24*QPochhammer[1/4, 1/4]) = 1/(24*A100221) = 0.060514735102066542326446... . - Vaclav Kotesovec, Jun 01 2015

cp's OEIS Frontend

A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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