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 A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n. 21
 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, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022 LINKS Alois P. Heinz, Rows n = 1..141, flattened FORMULA 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 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015 Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019 Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021 G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021 EXAMPLE Table begins: 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, 12, 1; 22, 141, 321, 362, 225, 78, 14, 1; 30, 240, 669, 944, 765, 370, 105, 16, 1; 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1; ... 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. MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, k*add( A(n-j, k)*numtheory[sigma](j), j=1..n)/n) end: T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k): seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Mar 12 2015 # Uses function PMatrix from A357368. Adds row and column for n, k = 0. PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022 MATHEMATICA 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 *) CROSSREFS Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236. Row sums give A055887. Cf. A097805, A010815, A055888, A144064, A261719, A326346. T(2n,n) gives A340987. Sequence in context: A126198 A055888 A094442 * A306186 A154929 A249042 Adjacent sequences: A060639 A060640 A060641 * A060643 A060644 A060645 KEYWORD easy,nonn,tabl AUTHOR Alford Arnold, Apr 16 2001 EXTENSIONS More terms from Vladeta Jovovic, Jan 02 2004 STATUS approved

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Last modified November 30 00:02 EST 2022. Contains 358431 sequences. (Running on oeis4.)