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A346160 a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.

%I A346160 #55 Apr 05 2024 08:20:58
%S A346160 0,25,1041,15655,637854,2507860,35577568
%N A346160 a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.
%C A346160 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 A346160 Stephen DeSalvo and Igor Pak, Log-concavity of the partition function, The Ramanujan Journal, 38 (2015), 61-73.
%H A346160 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 A346160 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 A346160 For n=0, P_0(k)^2 >= P_0(k-1)*P_0(k+1) for all k > 0.
%e A346160 For n=1, P_1(k)^2 >= P_1(k-1)*P_1(k+1) for all k > 24.
%e A346160 For n=2, P_2(k)^2 >= P_2(k-1)*P_2(k+1) for all k > 1042.
%e A346160 For n=3, P_3(k)^2 >= P_3(k-1)*P_3(k+1) for all k > 15656.
%e A346160 No further terms are known.
%o A346160 (SageMath)
%o A346160 def power_partition_generating_series(s, n=20):
%o A346160     R = ZZ['q']
%o A346160     ans = R.one()
%o A346160     m = 1
%o A346160     while m**s < n:
%o A346160         l = [0] * n
%o A346160         l[0] = 1
%o A346160         for i in range(1, (n + m**s - 1) // m**s):
%o A346160             l[i*m**s] = 1
%o A346160         ans = ans._mul_trunc_(R(l), n)
%o A346160         m += 1
%o A346160     return ans
%o A346160 %time
%o A346160 seq = power_partition_generating_series(2, 15000).coefficients()
%o A346160 last_neg = None
%o A346160 for n in range(2, len(seq) - 1):
%o A346160     d = seq[n]**2 / seq[n-1] / seq[n+1]
%o A346160     if d < 1:
%o A346160         last_neg = n
%o A346160 print(last_neg)
%o A346160 # _Vincent Delecroix_, Dec 28 2022
%Y A346160 Cf. A000041, A001156, A003108, A046042.
%K A346160 nonn,more
%O A346160 0,2
%A A346160 _Brennan G. Benfield_, Sep 28 2021
%E A346160 Data corrected by _Brennan G. Benfield_, Dec 28 2022