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 A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 11
 1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition (number theory) FORMULA Sum_{k=1..n} k * T(n,k) = A327557(n). EXAMPLE T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a. Triangle T(n,k) begins:   1;   0, 1;   0, 1,   2;   0, 2,   6,    5;   0, 2,  15,   27,    15;   0, 3,  32,  102,   124,     52;   0, 4,  65,  319,   656,    600,    203;   0, 5, 124,  897,  2780,   4210,   3084,    877;   0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;   0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;   ... MAPLE C:= binomial: b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))     end: T:= (n, k)-> add(b(n\$2, i)*(-1)^(k-i)*C(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA c = Binomial; b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]]; T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give: A000007, A000009 (for n>0), A327598. Main diagonal gives A000110. Row sums give A317776. T(2n,n) gives A327556. Cf. A255903, A326914, A326962, A327117, A327557, A309973. Sequence in context: A325199 A185197 A323845 * A157491 A094385 A291799 Adjacent sequences:  A327113 A327114 A327115 * A327117 A327118 A327119 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 13 2019 STATUS approved

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Last modified January 22 19:45 EST 2022. Contains 350504 sequences. (Running on oeis4.)