A320552 -id:A320552 - 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 *)

A258465 Number of partitions of n into parts of exactly 10 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 56, 1762, 41143, 795657, 13499449, 208050040, 2979881876, 40300054520, 520576172762, 6478447651345, 78185947269684, 919805200917658, 10591351937396242, 119764715367192468, 1333512940732309728, 14652754322423701707, 159182411488944508232

Offset: 10

Alois P. Heinz, May 30 2015

nonn

In general, column k>1 of A256130 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

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=10 of A256130.

Cf. A320552.

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,10):
seq(a(n), n=10..30);

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, 10], {n, 10, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) ~ c * 10^n, where c = 1/(10!*Product_{n>=1} (1-1/10^n)) = 1/(10!*QPochhammer[1/10, 1/10]) = 0.0000003096292864992979803727261336621564... . - 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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