A347588 -id:A347588

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A347586

Number of partitions of n into at most 4 distinct parts.

+10
8

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 36, 43, 49, 57, 65, 75, 84, 96, 107, 121, 134, 150, 165, 184, 201, 222, 242, 266, 288, 315, 340, 370, 398, 431, 462, 499, 533, 573, 611, 655, 696, 744, 789, 841, 890, 946, 999, 1060, 1117, 1182, 1244, 1314, 1380, 1455

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..60.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

FORMULA

G.f.: Sum_{k=0..4} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

a(n) ~ A000578(n)/144. - Stefano Spezia, Sep 08 2021

MATHEMATICA

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 4}], {x, 0, nmax}], x]

Join[{1}, LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 8, 10}, 60]]

CROSSREFS

Cf. A000578, A001400, A014591, A065033, A347587, A347588.

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Sep 08 2021

STATUS

approved

A347587

Number of partitions of n into at most 5 distinct parts.

+10
3

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 75, 88, 102, 119, 137, 158, 181, 207, 235, 268, 302, 341, 383, 430, 480, 536, 595, 661, 731, 808, 889, 979, 1073, 1176, 1285, 1403, 1527, 1662, 1803, 1956, 2116, 2288, 2468, 2662, 2864, 3080, 3306, 3547

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..58.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

FORMULA

G.f.: Sum_{k=0..5} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

MATHEMATICA

nmax = 58; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 5}], {x, 0, nmax}], x]

LinearRecurrence[{1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1}, {1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22}, 59]

CROSSREFS

Cf. A001401, A014591, A065033, A347586, A347588.

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Sep 08 2021

STATUS

approved

A358011

Number of partitions of n into at most 6 distinct prime parts.

+10
2

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 31, 35, 36, 39, 40, 42, 42, 45, 49, 50, 52, 55, 53, 61, 61, 67, 67, 70, 70, 77, 77, 86, 84

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..80.

CROSSREFS

Cf. A000040, A000586, A219180, A219200, A223893, A347550, A347588, A347610, A347743, A358009, A358010.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 24 2022

STATUS

approved

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