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%I #38 Oct 16 2022 03:24:47
%S 1,2,1,3,4,1,5,10,6,1,7,22,21,8,1,11,43,59,36,10,1,15,80,144,124,55,
%T 12,1,22,141,321,362,225,78,14,1,30,240,669,944,765,370,105,16,1,42,
%U 397,1323,2266,2287,1437,567,136,18,1,56,640,2511,5100,6215,4848,2478,824,171,20,1
%N Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.
%C Also the convolution triangle of A000041. - _Peter Luschny_, Oct 07 2022
%H Alois P. Heinz, <a href="/A060642/b060642.txt">Rows n = 1..141, flattened</a>
%F G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - _Vladeta Jovovic_, Jan 02 2004
%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - _Alois P. Heinz_, Mar 12 2015
%F Sum_{k=1..n} k * T(n,k) = A326346(n). - _Alois P. Heinz_, Sep 11 2019
%F Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - _Alois P. Heinz_, Feb 07 2021
%F G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - _Ilya Gutkovskiy_, Feb 13 2021
%e Table begins:
%e 1;
%e 2, 1;
%e 3, 4, 1;
%e 5, 10, 6, 1;
%e 7, 22, 21, 8, 1;
%e 11, 43, 59, 36, 10, 1;
%e 15, 80, 144, 124, 55, 12, 1;
%e 22, 141, 321, 362, 225, 78, 14, 1;
%e 30, 240, 669, 944, 765, 370, 105, 16, 1;
%e 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
%e ...
%e For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.
%p A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
%p A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
%p end:
%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, Mar 12 2015
%p # Uses function PMatrix from A357368. Adds row and column for n, k = 0.
%p PMatrix(10, combinat:-numbpart); # _Peter Luschny_, Oct 07 2022
%t A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Jul 15 2015, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236.
%Y Row sums give A055887.
%Y Cf. A097805, A010815, A055888, A144064, A261719, A326346.
%Y T(2n,n) gives A340987.
%K easy,nonn,tabl
%O 1,2
%A _Alford Arnold_, Apr 16 2001
%E More terms from _Vladeta Jovovic_, Jan 02 2004
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