A060850 - OEIS

A060850

Array of the coefficients A(n,k) in the expansion of Product_{i>=1} 1/(1-x^i)^n = Sum_{k>=0} A(n,k)*x^k, n >= 1, k >= 0.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 7, 1, 6, 20, 40, 51, 36, 11, 1, 7, 27, 65, 105, 108, 65, 15, 1, 8, 35, 98, 190, 252, 221, 110, 22, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 1, 11, 65, 255

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OFFSET

1,5

COMMENTS

Table read by antidiagonals: entry (n,k) gives number of partitions of n objects into parts of k kinds. - Franklin T. Adams-Watters, Dec 28 2006

LINKS

Table of n, a(n) for n=1..70.

FORMULA

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=1..n} A000041(k-1)*A(n-k;x)*x^(k-1), A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004

EXAMPLE

Table (row k, k >= 0: number of partitions of n, n >= 0, into parts of k kinds):

Array begins:

=======================================================================

k\n| 0 1 2 3 4 5 6 7 8 9 10

---|-------------------------------------------------------------------

1 | 1 1 2 3 5 7 11 15 22 30 42

2 | 1 2 5 10 20 36 65 110 185 300 481

3 | 1 3 9 22 51 108 221 429 810 1479 2640

4 | 1 4 14 40 105 252 574 1240 2580 5180 10108

5 | 1 5 20 65 190 506 1265 2990 6765 14725 31027

6 | 1 6 27 98 315 918 2492 6372 15525 36280 81816

7 | 1 7 35 140 490 1547 4522 12405 32305 80465 192899

8 | 1 8 44 192 726 2464 7704 22528 62337 164560 417140

9 | 1 9 54 255 1035 3753 12483 38709 113265 315445 841842

10 | 1 10 65 330 1430 5512 19415 63570 195910 573430 1605340

11 | 1 11 77 418 1925 7854 29183 100529 325193 997150 2919411

...

Triangle (row n, n >= 0: number of partitions of n into parts of n - k kinds, 0 <= k <= n) (antidiagonals of above table) (parenthesized last term on each row, which would correspond to row k = 0 in above table)

Triangle begins: (column k: n - k kinds of parts)

===================================

n\k| 0 1 2 3 4 5 6 7

---+-------------------------------

0 |(1)

1 | 1, (0)

2 | 1, 1, (0)

3 | 1, 2, 2, (0)

4 | 1, 3, 5, 3, (0)

5 | 1, 4, 9, 10, 5, (0)

6 | 1, 5, 14, 22, 20, 7, (0)

7 | 1, 6, 20, 40, 51, 36, 11, (0)

...

MATHEMATICA

t[n_, k_] := CoefficientList[ Series[ Product[1/(1 - x^i)^n, {i, k}], {x, 0, k}], x][[k]]; (* Robert G. Wilson v, Aug 08 2018 *)

t[n_, k_]; = IntegerPartitions[n, {k}]; Table[ t[n - k + 1, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 08 2018 *)

CROSSREFS

Cf. A067687 (table antidiagonal sums, triangle row sums).

Rows (table), diagonals (triangle): A000041, A000712, A000716, A023003-A023021, A006922.

Columns (table, triangle): A000012, A001477, A000096, A006503, A006504.

Sequence in context: A086350 A239830 A140767 * A208336 A038137 A073133

Adjacent sequences: A060847 A060848 A060849 * A060851 A060852 A060853

KEYWORD

tabl,nonn,easy

AUTHOR

Bo T. Ahlander (ahlboa(AT)isk.kth.se), May 03 2001

EXTENSIONS

More terms from Vladeta Jovovic, Jan 02 2004

STATUS

approved