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 A326914 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 distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows. 9
 1, 1, 2, 2, 5, 1, 12, 15, 18, 64, 52, 20, 166, 340, 203, 18, 332, 1315, 1866, 877, 15, 566, 3895, 9930, 10710, 4140, 11, 864, 9770, 39960, 74438, 64520, 21147, 6, 1214, 21848, 134871, 386589, 564508, 408096, 115975, 3, 1596, 44880, 402756, 1668338, 3652712 (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, Rows n = 0..200, flattened Wikipedia, Partition (number theory) FORMULA Sum_{k=1..n} k * T(n,k) = A327115(n). T(n*2^(n-1),n) = T(A001787(n),n) = 1. T(n*2^(n-1)-1,n) = n for n >= 2. EXAMPLE T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c. Triangle T(n,k) begins:   1;      1;         2;         2,  5;         1, 12,   15;            18,   64,    52;            20,  166,   340,    203;            18,  332,  1315,   1866,    877;            15,  566,  3895,   9930,  10710,   4140;            11,  864,  9770,  39960,  74438,  64520,  21147;             6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;   ... 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) 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), 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=h(n)..n), n=0..12); 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_] := 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], j], {j, 0, n/i}]]]; T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *) CROSSREFS Main diagonal gives A000110. Row sums give A116539. Column sums give A003465. Cf. A001787, A255903, A326962 (this triangle read by columns), A327115, A327116, A327117. Sequence in context: A326616 A249033 A068762 * A155679 A319771 A021448 Adjacent sequences:  A326911 A326912 A326913 * A326915 A326916 A326917 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Sep 13 2019 STATUS approved

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Last modified January 16 15:53 EST 2021. Contains 340206 sequences. (Running on oeis4.)