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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

EXAMPLE

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;

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Jan 24 2017, translated from Maple *)

CROSSREFS

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.

Sequence in context: A296207 A253628 A102728 * A323174 A295683 A165519

Adjacent sequences:  A262492 A262493 A262494 * A262496 A262497 A262498

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 24 2015

STATUS

approved

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Last modified January 29 07:03 EST 2020. Contains 331337 sequences. (Running on oeis4.)