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

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

Offset: 0

Author

Brennan G. Benfield, Sep 28 2021

Keywords

• nonn
• more

Comments

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.

Examples

			For n=0, P_0(k)^2 >= P_0(k-1)*P_0(k+1) for all k > 0.
For n=1, P_1(k)^2 >= P_1(k-1)*P_1(k+1) for all k > 24.
For n=2, P_2(k)^2 >= P_2(k-1)*P_2(k+1) for all k > 1042.
For n=3, P_3(k)^2 >= P_3(k-1)*P_3(k+1) for all k > 15656.
No further terms are known.
		

References

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

Links

• Brennan Benfield and Arindam Roy, Log-concavity And The Multiplicative Properties of Restricted Partition Functions, arXiv:2404.03153 [math.NT], 2024.
• Stephen DeSalvo and Igor Pak, Log-Concavity of the Partition Function, arXiv:1310.7982 [math.CO], 2013-2014.

Crossrefs

Cf. A000041, A001156, A003108, A046042.

Programs

SageMath

def power_partition_generating_series(s, n=20):
    R = ZZ['q']
    ans = R.one()
    m = 1
    while m**s < n:
        l = [0] * n
        l[0] = 1
        for i in range(1, (n + m**s - 1) // m**s):
            l[i*m**s] = 1
        ans = ans.mul_trunc(R(l), n)
        m += 1
    return ans
%time
seq = power_partition_generating_series(2, 15000).coefficients()
last_neg = None
for n in range(2, len(seq) - 1):
    d = seq[n]**2 / seq[n-1] / seq[n+1]
    if d < 1:
        last_neg = n
print(last_neg)
# Vincent Delecroix, Dec 28 2022

Extensions

Data corrected by Brennan G. Benfield, Dec 28 2022