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A360674
Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.
8
1, 1, 3, 4, 7, 6, 12, 9, 16, 15, 21, 16, 34, 22, 33, 36, 47, 36, 62, 44, 75, 68, 78, 68, 120, 93, 113, 117, 151, 122, 195, 148, 209, 197, 220, 226, 315, 249, 304, 309, 402, 332, 463, 387, 496, 515, 539, 514, 712, 609, 738, 723, 845, 774, 983, 914, 1111
OFFSET
0,3
COMMENTS
Of course, only one of the two conditions is necessary.
FORMULA
a(n) = A360672(2n,n).
EXAMPLE
The a(1) = 1 through a(6) = 12 partitions:
(11) (22) (33) (44) (55) (66)
(211) (321) (422) (532) (633)
(1111) (21111) (431) (541) (642)
(111111) (2222) (32221) (651)
(22211) (211111111) (3333)
(2111111) (1111111111) (33222)
(11111111) (33321)
(42222)
(222222)
(2222211)
(21111111111)
(111111111111)
For example, the partition y = (3,2,2,2,1) has halves (3,2) and (2,2,1), both with sum 5, so y is counted under a(5).
MATHEMATICA
Table[Length[Select[IntegerPartitions[2n], Total[Take[#, Floor[Length[#]/2]]]==n&]], {n, 0, 15}]
PROG
(Python)
def accel_asc(n):
a = [0 for i in range(n + 1)]
k = 1
y = n - 1
while k != 0:
x = a[k - 1] + 1
k -= 1
while 2 * x <= y:
a[k] = x
y -= x
k += 1
l = k + 1
while x <= y:
a[k] = x
a[l] = y
yield a[:k + 2]
x += 1
y -= 1
a[k] = x + y
y = x + y - 1
yield a[:k + 1]
for y in range(1000):
num = 0
for x in accel_asc(2*y):
stop = len(x)//2+1
if len(x) % 2 == 0:
stop -= 1
right = x[0:stop]
left = x[stop:]
if sum(right) == sum(left):
num += 1
print(y, num)
# David Consiglio, Jr., Mar 09 2023
CROSSREFS
The even-length case is A000005.
Central diagonal of A360672.
These partitions have ranks A360953.
A008284 counts partitions by length, row sums A000041.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Sequence in context: A348944 A369761 A003981 * A324108 A324054 A214413
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 04 2023
EXTENSIONS
More terms from David Consiglio, Jr., Mar 09 2023
STATUS
approved