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 A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 9
 1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 18, 15, 0, 1, 10, 45, 84, 52, 0, 1, 14, 94, 298, 415, 203, 0, 1, 18, 174, 844, 1995, 2178, 877, 0, 1, 23, 300, 2081, 7440, 13638, 12131, 4140, 0, 1, 28, 486, 4652, 23670, 64898, 95823, 71536, 21147, 0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The sequence of column k satisfies a linear recurrence with constant coefficients of order k*2^(k-1) = A001787(k). LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition (number theory) FORMULA Sum_{k=1..n} k * T(n,k) = A327118(n). EXAMPLE T(3,2) = 4: 2ab1a, 2ab1b, 1a1a1b, 1a1b1b. Triangle T(n,k) begins:   1;   0, 1;   0, 1,  2;   0, 1,  4,   5;   0, 1,  7,  18,   15;   0, 1, 10,  45,   84,    52;   0, 1, 14,  94,  298,   415,    203;   0, 1, 18, 174,  844,  1995,   2178,    877;   0, 1, 23, 300, 2081,  7440,  13638,  12131,   4140;   0, 1, 28, 486, 4652, 23670,  64898,  95823,  71536,  21147;   0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975;   ... 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)+j-1, 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 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] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]]; T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *) CROSSREFS Columns k=0-3 give: A000007, A057427, A014616(n-1) for n>1, A327842. Main diagonal gives A000110. Row sums give A116540. T(2n,n) gives A327843. Cf. A001787, A255903, A326914, A326962, A327116, A327118. Sequence in context: A247126 A342134 A349740 * A229223 A128749 A106579 Adjacent sequences:  A327114 A327115 A327116 * A327118 A327119 A327120 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 13 2019 STATUS approved

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Last modified January 24 19:03 EST 2022. Contains 350565 sequences. (Running on oeis4.)