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A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n. 0
1, 3, 1, 6, 1, 2, 12, 1, 4, 3, 20, 1, 4, 5, 5, 35, 1, 6, 8, 9, 7, 54, 1, 6, 10, 12, 11, 11, 86, 1, 8, 13, 20, 14, 19, 15, 128, 1, 8, 18, 23, 22, 25, 23, 22, 192, 1, 10, 21, 34, 30, 37, 29, 36, 30, 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42, 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the construction of the diagram see A225600.

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.

.

.                       Illustration of the parts

.    Diagram             that start in column k:

.    for n=5       k=1          k=2  k=3    k=4    k=5

.   _ _ _ _ _       _ _ _ _ _

.  |_ _ _    |     |_ _ _ _ _|               _ _

.  |_ _ _|_  |     |_ _ _|_                 |_ _|   _

.  |_ _    | |     |_ _ _ _|          _ _          |_|

.  |_ _|_  | |     |_ _|_            |_ _|   _     |_|

.  |_ _  | | |     |_ _ _|            _     |_|    |_|

.  |_  | | | |     |_ _|         _   |_|    |_|    |_|

.  |_|_|_|_|_|     |_|          |_|  |_|    |_|    |_|

.

k = 1 2 3 4 5

.

The 5th row lists:  20           1    4      5      5

.

Triangle begins:

1;

3,   1;

6,   1,  2;

12,  1,  4,  3;

20,  1,  4,  5,  5;

35,  1,  6,  8,  9,  7;

54,  1,  6, 10, 12, 11, 11;

86,  1,  8, 13, 20, 14, 19, 15;

128, 1,  8, 18, 23, 22, 25, 23, 22;

192, 1, 10, 21, 34, 30, 37, 29, 36, 30;

275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;

399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;

CROSSREFS

Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.

Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.

Sequence in context: A171852 A010287 A130270 * A090049 A115364 A016476

Adjacent sequences:  A225596 A225597 A225598 * A225600 A225601 A225602

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Aug 02 2013

STATUS

approved

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Last modified June 20 10:13 EDT 2019. Contains 324234 sequences. (Running on oeis4.)