A389679 - OEIS

A389679

Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions with k designated summands of all positive integers <= n, with n >= 1, 1 <= k <= A003056(n).

1, 6, 16, 2, 37, 10, 63, 38, 113, 96, 3, 163, 224, 14, 248, 446, 51, 339, 818, 151, 469, 1414, 367, 4, 591, 2310, 818, 18, 801, 3588, 1666, 64, 971, 5396, 3177, 186, 1221, 7918, 5670, 488, 1481, 11162, 9791, 1114, 5, 1822, 15554, 16092, 2398, 22, 2112, 21076, 25714, 4800, 77

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OFFSET

1,2

COMMENTS

When part i has multiplicity j > 0 exactly one part i is "designated".

Row n has length A003056(n), hence the first positive element in column k is in the row A000217(k).

Column k gives the partial sums of the column k of A388063.

LINKS

Table of n, a(n) for n=1..55.

G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.

William Y. C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y. Y. Shen, On the Number of Partitions with Designated Summands, arXiv:1208.2210 [math.CO], 2012.

EXAMPLE

Triangle begins:

------------------------------------

n\k: 1 2 3 4 5

------------------------------------

1 | 1;

2 | 6;

3 | 16, 2;

4 | 37, 10;

5 | 63, 38;

6 | 113, 96, 3;

7 | 163, 224, 14;

8 | 248, 446, 51;

9 | 339, 818, 151;

10 | 469, 1414, 367, 4;

11 | 591, 2310, 818, 18;

12 | 801, 3588, 1666, 64;

13 | 971, 5396, 3177, 186;

14 | 1221, 7918, 5670, 488;

15 | 1481, 11162, 9791, 1114, 5;

16 | 1822, 15554, 16092, 2398, 22;

17 | 2112, 21076, 25714, 4800, 77;

...

CROSSREFS

Column 1 gives A064602.

Cf. A000217, A001157, A003056, A077285, A384999, A385001, A386997, A388063.

Sequence in context: A225304 A299709 A107777 * A136140 A155834 A085452

Adjacent sequences: A389676 A389677 A389678 * A389680 A389681 A389682

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Oct 10 2025

STATUS

approved