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A327584 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns. 5
1, 1, 3, 4, 6, 13, 48, 150, 300, 666, 936, 1824, 2520, 2160, 5040, 75, 536, 2820, 11144, 41346, 131304, 420084, 1191552, 3427008, 9207456, 23466720, 61522560, 141553560, 345346560, 777152160, 1635096960, 3700806480, 6998261760, 14211912960, 27442437120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.

LINKS

Alois P. Heinz, Columns k = 0..7, flattened

EXAMPLE

T(3,2) = 4: 2ab1a, 2ab1b, 1a2ab, 1b2ab.

T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.

T(4,2) = 6: 2ab1a1b, 1a2ab1b, 1a1b2ab, 2ab1b1a, 1b2ab1a, 1b1a2ab.

Triangle T(n,k) begins:

  1;

     1;

        3;

        4,  13;

        6,  48,    75;

           150,   536,    541;

           300,  2820,   6320,   4683;

           666, 11144,  50150,  81012,   47293;

           936, 41346, 308080, 903210, 1134952, 545835;

           ...

MAPLE

C:= binomial:

g:= proc(n) option remember; n*2^(n-1) end:

h:= proc(n) option remember; local k; for k from

      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od

    end:

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

      `if`(i<1 or k<h(n), 0, add(b(n-i*j, min(n-i*j, i-1),

        k, p+j)*C(C(k, i), j), j=0..n/i)))

    end:

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

seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);

MATHEMATICA

c = Binomial;

g[n_] := g[n] = n*2^(n - 1);

h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0,

     h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];

b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!,

     If[i < 1 || k < h[n], 0, Sum[b[n - i*j, Min[n - i*j, i - 1],

     k, p + j]*c[c[k, i], j], {j, 0, n/i}]]];

T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k - i)*c[k, i], {i, 0, k}];

Table[Table[T[n, k], {n, k, k*2^(k - 1)}], {k, 0, 5}] // Flatten (* Jean-Fran├žois Alcover, Feb 22 2021, after Alois P. Heinz *)

CROSSREFS

Main diagonal gives A000670.

Row sums give A321587.

Column sums give A327585.

Cf. A001787, A326962, A327583 (this triangle read by rows).

Sequence in context: A180647 A318345 A143100 * A180859 A271618 A137820

Adjacent sequences:  A327581 A327582 A327583 * A327585 A327586 A327587

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 17 2019

STATUS

approved

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Last modified January 23 03:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)