A326617 - OEIS

A326617

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 distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

1, 1, 2, 2, 1, 5, 9, 9, 10, 9, 3, 13, 44, 96, 152, 155, 124, 140, 160, 113, 48, 16, 4, 42, 225, 680, 1350, 2180, 3751, 6050, 7420, 6870, 5555, 5330, 6300, 6475, 5025, 3000, 1250, 250, 150, 1098, 4155, 11730, 30300, 69042, 127364, 188568, 249690, 365160, 584733

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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, Columns k = 0..40, flattened

Wikipedia, Partition (number theory)

FORMULA

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

EXAMPLE

T(3,2) = 2: 2a1b, 2b1a.

T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.

Triangle T(n,k) begins:

2, 5;

1, 9, 13;

9, 44, 42;

10, 96, 225, 150;

9, 152, 680, 1098, 576;

3, 155, 1350, 4155, 5201, 2266;

124, 2180, 11730, 26642, 26904, 9966;

140, 3751, 30300, 106281, 182000, 149832, 47466;

...

MAPLE

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:

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))(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), n=k..g(k)), k=0..6);

MATHEMATICA

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];

T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];

Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

CROSSREFS

Main diagonal gives A178682.

Row sums give A326648.

Column sums give A326650.

Cf. A000203, A024916, A326616 (this triangle read by rows), A326649, A326651.

Sequence in context: A342722 A344528 A329429 * A371727 A326962 A350820

Adjacent sequences: A326614 A326615 A326616 * A326618 A326619 A326620

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 12 2019

STATUS

approved