%I #52 Oct 19 2022 10:48:49

%S 1,0,1,0,1,1,0,2,2,1,0,2,5,3,1,0,3,8,9,4,1,0,4,14,19,14,5,1,0,5,22,39,

%T 36,20,6,1,0,6,34,72,85,60,27,7,1,0,8,50,128,180,160,92,35,8,1,0,10,

%U 73,216,360,381,273,133,44,9,1,0,12,104,354,680,845,720,434,184,54,10,1

%N Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%C For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019

%C T is the convolution triangle of A000009 (see A357368). - _Peter Luschny_, Oct 19 2022

%H Alois P. Heinz, <a href="/A308680/b308680.txt">Rows n = 0..200, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).

%F Sum_{k=1..n} k * T(n,k) = A325915(n).

%F G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - _Alois P. Heinz_, Jan 29 2021

%e T(4,1) = 2: 3a1a, 4a.

%e T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.

%e T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.

%e T(4,4) = 1: 1a1b1c1d.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 2, 1;

%e 0, 2, 5, 3, 1;

%e 0, 3, 8, 9, 4, 1;

%e 0, 4, 14, 19, 14, 5, 1;

%e 0, 5, 22, 39, 36, 20, 6, 1;

%e 0, 6, 34, 72, 85, 60, 27, 7, 1;

%e 0, 8, 50, 128, 180, 160, 92, 35, 8, 1;

%e 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;

%e ...

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->

%p b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))

%p end:

%p T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..12);

%p # second Maple program:

%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

%p `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p end:

%p T:= proc(n, k) option remember;

%p `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),

%p (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Jan 31 2021

%p # Uses function PMatrix from A357368.

%p PMatrix(10, A000009); # _Peter Luschny_, Oct 19 2022

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

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

%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, from Maple *)

%Y Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.

%Y Main diagonal and lower diagonals give: A000012, A001477, A000096.

%Y Row sums give A304969.

%Y T(2n,n) gives A324595.

%Y Cf. A060642, A286335, A325915.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Aug 29 2019