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A327245 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 colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 8
1, 0, 1, 0, 1, 3, 0, 3, 10, 13, 0, 3, 39, 87, 75, 0, 5, 100, 510, 836, 541, 0, 11, 303, 2272, 7042, 9025, 4683, 0, 13, 782, 9999, 46628, 104255, 109110, 47293, 0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835, 0, 27, 5388, 154038, 1577256, 7676830, 19798096, 28538496, 21713032, 7087261 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

FORMULA

Sum_{k=1..n} k * T(n,k) = A327588(n).

EXAMPLE

T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.

T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.

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

Triangle T(n,k) begins:

  1;

  0,  1;

  0,  1,    3;

  0,  3,   10,    13;

  0,  3,   39,    87,     75;

  0,  5,  100,   510,    836,    541;

  0, 11,  303,  2272,   7042,   9025,    4683;

  0, 13,  782,  9999,  46628, 104255,  109110,   47293;

  0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835;

  ...

MAPLE

C:= binomial:

b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(

      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, 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), k=0..n), n=0..10);

MATHEMATICA

c = Binomial;

b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p + j] c[c[k + i - 1, 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[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 29 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-2 give: A000007, A032020 (for n>0), A327847.

Main diagonal gives A000670.

Row sums give A321586.

T(2n,n) gives A327589.

Cf. A327244, A327588.

Sequence in context: A177785 A212036 A191619 * A157525 A157521 A176005

Adjacent sequences:  A327242 A327243 A327244 * A327246 A327247 A327248

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 14 2019

STATUS

approved

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Last modified November 29 01:40 EST 2021. Contains 349416 sequences. (Running on oeis4.)