A388063 - OEIS

A388063

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

1, 5, 10, 2, 21, 8, 26, 28, 50, 58, 3, 50, 128, 11, 85, 222, 37, 91, 372, 100, 130, 596, 216, 4, 122, 896, 451, 14, 210, 1278, 848, 46, 170, 1808, 1511, 122, 250, 2522, 2493, 302, 260, 3244, 4121, 626, 5, 341, 4392, 6301, 1284, 17, 290, 5522, 9622, 2402, 55

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OFFSET

1,2

COMMENTS

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

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:

10, 2;

21, 8;

26, 28;

50, 58, 3;

50, 128, 11;

85, 222, 37;

91, 372, 100;

130, 596, 216, 4;

122, 896, 451, 14;

210, 1278, 848, 46;

170, 1808, 1511, 122;

250, 2522, 2493, 302;

260, 3244, 4121, 626, 5;

341, 4392, 6301, 1284, 17;

290, 5522, 9622, 2402, 55;

...

For n = 4 and k = 1 there are 21 parts in all partitions of 4 with only one designated summand. They are [4'], [2'+ 2], [2 + 2'], [1'+ 1 + 1 + 1], [1 + 1'+ 1 + 1], [1 + 1 + 1'+ 1], [1 + 1 + 1 + 1'], so T(4,1) = 21.

For n = 4 and k = 2 there are 8 parts in all partitions of 4 with two designated summands. They are [3'+ 1'], [2'+ 1'+ 1], [2'+ 1 + 1'], so T(4,2) = 8.

CROSSREFS

Column 1 gives A001157.

Row sums give A388064.

Cf. A000217, A002127, A002128, A003056, A006128, A066186, A077285, A384998, A384999, A293421, A365664, A365665, A384926, A384998, A384999, A385001, A386997, A386998, A388064.

Sequence in context: A361672 A169841 A084341 * A054513 A295121 A066200

Adjacent sequences: A388060 A388061 A388062 * A388064 A388065 A388066

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Sep 14 2025

EXTENSIONS

More terms from Sean A. Irvine, Sep 21 2025

STATUS

approved