A309880 - OEIS

A309880

Sum of the odd parts appearing among the fourth largest parts in the partitions of n into 5 parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 18, 26, 34, 46, 57, 72, 87, 110, 133, 165, 201, 246, 291, 349, 407, 481, 559, 653, 754, 875, 1003, 1154, 1309, 1496, 1690, 1913, 2152, 2423, 2707, 3032, 3373, 3763, 4169, 4627, 5109, 5643, 6204, 6825, 7473, 8197

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OFFSET

0,8

LINKS

Table of n, a(n) for n=0..53.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1).

FORMULA

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} k * (k mod 2).

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.

EXAMPLE

Figure 1: The partitions of n into 5 parts for n = 10, 11, ..

1+1+1+1+10

1+1+1+2+9

1+1+1+3+8

1+1+1+4+7

1+1+1+5+6

1+1+1+1+9 1+1+2+2+8

1+1+1+2+8 1+1+2+3+7

1+1+1+3+7 1+1+2+4+6

1+1+1+4+6 1+1+2+5+5

1+1+1+5+5 1+1+3+3+6

1+1+1+1+8 1+1+2+2+7 1+1+3+4+5

1+1+1+2+7 1+1+2+3+6 1+1+4+4+4

1+1+1+3+6 1+1+2+4+5 1+2+2+2+7

1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6

1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5

1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5

1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4

1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4

1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6

1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5

1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4

1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4

2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3

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n | 10 11 12 13 14 ...

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a(n) | 5 7 8 13 18 ...

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MATHEMATICA

LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 18, 26, 34, 46, 57, 72, 87, 110, 133, 165, 201, 246, 291, 349, 407, 481, 559, 653, 754, 875, 1003, 1154, 1309}, 50]

Table[Total[Select[IntegerPartitions[n, {5}][[;; , 4]], OddQ]], {n, 0, 60}] (* Harvey P. Dale, Oct 10 2024 *)

CROSSREFS

Cf. A309879, A309881, A309882.

Sequence in context: A124168 A309708 A285929 * A054762 A039088 A111794

Adjacent sequences: A309877 A309878 A309879 * A309881 A309882 A309883

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Aug 21 2019

STATUS

approved