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 A326500 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 5
 1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 22, 30, 13, 0, 7, 54, 129, 124, 42, 0, 11, 118, 428, 696, 525, 150, 0, 15, 248, 1293, 3108, 3830, 2358, 576, 0, 22, 490, 3483, 11595, 20720, 20535, 10661, 2266, 0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Rows n = 1..140, flattened Wikipedia, Partition (number theory) FORMULA Sum_{k=1..n} k * T(n,k) = A326656(n). EXAMPLE T(3,1) = 3: 3aaa, 2aa1a, 111aaa. T(3,2) = 8: 3aab, 3abb, 2aa1b, 2ab1b, 2ab1a, 2bb1a, 111aab, 111abb. T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc. Triangle T(n,k) begins:   1;   0,  1;   0,  2,   2;   0,  3,   8,    5;   0,  5,  22,   30,    13;   0,  7,  54,  129,   124,    42;   0, 11, 118,  428,   696,   525,    150;   0, 15, 248, 1293,  3108,  3830,   2358,    576;   0, 22, 490, 3483, 11595, 20720,  20535,  10661,  2266;   0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966;   ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->       b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))     end: T:= (n, k)-> add(b(n\$2, k-i)*(-1)^i*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=0..12); CROSSREFS Columns k=0-1 give: A000007, A000041 (for n>0). Main diagonal gives A178682. Row sums give A326654. T(2n,n) gives A328158. Cf. A326656. Sequence in context: A061314 A193383 A218033 * A255903 A118262 A065484 Adjacent sequences:  A326497 A326498 A326499 * A326501 A326502 A326503 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 12 2019 STATUS approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)