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a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.
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%I #55 Apr 05 2024 08:20:58

%S 0,25,1041,15655,637854,2507860,35577568

%N a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.

%C The power partition function P_n(k) is a restriction on the partition function. P_n(k) equals the number of ways a positive integer k can be written as the sum of perfect n powers. DeSalvo and Pak showed that the partition function (P_1(k)) is log-concave for all k > 24.

%D Stephen DeSalvo and Igor Pak, Log-concavity of the partition function, The Ramanujan Journal, 38 (2015), 61-73.

%H Brennan Benfield and Arindam Roy, <a href="https://arxiv.org/abs/2404.03153">Log-concavity And The Multiplicative Properties of Restricted Partition Functions</a>, arXiv:2404.03153 [math.NT], 2024.

%H Stephen DeSalvo and Igor Pak, <a href="https://arxiv.org/abs/1310.7982">Log-Concavity of the Partition Function</a>, arXiv:1310.7982 [math.CO], 2013-2014.

%e For n=0, P_0(k)^2 >= P_0(k-1)*P_0(k+1) for all k > 0.

%e For n=1, P_1(k)^2 >= P_1(k-1)*P_1(k+1) for all k > 24.

%e For n=2, P_2(k)^2 >= P_2(k-1)*P_2(k+1) for all k > 1042.

%e For n=3, P_3(k)^2 >= P_3(k-1)*P_3(k+1) for all k > 15656.

%e No further terms are known.

%o (SageMath)

%o def power_partition_generating_series(s, n=20):

%o R = ZZ['q']

%o ans = R.one()

%o m = 1

%o while m**s < n:

%o l = [0] * n

%o l[0] = 1

%o for i in range(1, (n + m**s - 1) // m**s):

%o l[i*m**s] = 1

%o ans = ans._mul_trunc_(R(l), n)

%o m += 1

%o return ans

%o %time

%o seq = power_partition_generating_series(2, 15000).coefficients()

%o last_neg = None

%o for n in range(2, len(seq) - 1):

%o d = seq[n]**2 / seq[n-1] / seq[n+1]

%o if d < 1:

%o last_neg = n

%o print(last_neg)

%o # _Vincent Delecroix_, Dec 28 2022

%Y Cf. A000041, A001156, A003108, A046042.

%K nonn,more

%O 0,2

%A _Brennan G. Benfield_, Sep 28 2021

%E Data corrected by _Brennan G. Benfield_, Dec 28 2022