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A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n. 7
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

Row sums give A055887.

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

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

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-Fran├žois Alcover, Jul 15 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A048574, A055887, A055888, A144064, A261719.

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 June 16 11:12 EDT 2019. Contains 324152 sequences. (Running on oeis4.)