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A302300
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a(n) = Sum_{p in P} (Sum_{k_j = 1} 1)^2, where P is the set of partitions of n, and the k_j are the frequencies in p.
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2
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0, 1, 1, 5, 6, 12, 21, 33, 50, 79, 116, 169, 246, 346, 487, 675, 927, 1254, 1702, 2263, 3014, 3966, 5210, 6766, 8795, 11303, 14531, 18521, 23583, 29803, 37654, 47231, 59206, 73792, 91867, 113778, 140788, 173377, 213289, 261318, 319764, 389846, 474745, 576164
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OFFSET
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0,4
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COMMENTS
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This sequence is part of the contribution to the b^2 term of C_{1-b,2}(q) for(1-b,2)-colored partitions - partitions in which we can label parts any of an indeterminate 1-b colors, but are restricted to using only 2 of the colors per part size. This formula is known to match the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two up to the linear term in b.
It is of interest to enumerate and determine specific characteristics of partitions of n, considering each partition individually.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..4000
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398 [math.CO], 2008.
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.
W. J. Keith, Restricted k-color partitions, Ramanujan Journal (2016) 40: 71.
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FORMULA
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a(n) = Sum_{p in P} (Sum_{k_j = 1} 1)^2, where P is the set of partitions of n, and k_j are the frequencies in p.
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EXAMPLE
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For a(6), we sum over partitions of six. For each partition, we count 1 for each part which appears once, then square the total in each partition.
6............1^2 = 1
5,1..........2^2 = 4
4,2..........2^2 = 4
4,1,1........1^2 = 1
3,3..........0^2 = 0
3,2,1........3^2 = 9
3,1,1,1......1^2 = 1
2,2,2........0^2 = 0
2,2,1,1......0^2 = 0
2,1,1,1,1....1^2 = 1
1,1,1,1,1,1..0^2 = 0
--------------------
Total.............21
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MAPLE
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b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (
`if`(n=1, 1, 0)+p)^2, add(b(n-i*j, i-1,
`if`(j=1, 1, 0)+p), j=0..n/i))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Apr 05 2018
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MATHEMATICA
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Array[Total@ Map[Count[Split@ #, _?(Length@ # == 1 &)]^2 &, IntegerPartitions[#]] &, 43] (* Michael De Vlieger, Apr 05 2018 *)
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PROG
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(Python)
def frequencies(partition, n):
tot = 0
freq_list = []
i = 0
for p in partition:
freq = [0 for i in range(n+1)]
for i in p:
freq[i] += 1
for f in freq:
if f == 0:
tot += 1
freq_list.append(freq)
return freq_list
def sum_square_freqs_of_one(freq_part):
tot = 0
for f in freq_part:
count = 0
for i in f:
if i == 1:
count += 1
tot += count*count
return tot
#Pass in whichever value here for i, or iterate over the block of code
part = partitions(n)
freq_part = frequencies(part, n)
sum_of_ones = sum_square_freqs_of_one(freq_part)
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CROSSREFS
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Cf. A024786, A197126.
Sequence in context: A276407 A022310 A126593 * A173074 A099473 A051572
Adjacent sequences: A302297 A302298 A302299 * A302301 A302302 A302303
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KEYWORD
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nonn
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AUTHOR
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Emily Anible, Apr 04 2018
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STATUS
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approved
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