A024940 -id:A024940 - cp's OEIS frontend

A369787 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k*(k+1)/2)).

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 13, 19, 28, 44, 71, 112, 168, 249, 321, 419, 549, 652, 797, 939, 1104, 1265, 1440, 1638, 1842, 2059, 2295, 2538, 2809, 3087, 3385, 3698, 4032, 4381, 4754, 5143, 5554, 5985, 6437, 6910, 7405, 7922, 8463, 9027, 9615, 10227, 10865, 11528

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Cf. A024940, A039822.

a(n) = #Set(Vec(prod(k=1, n, 1+x^(k*(k+1)/2))));

from collections import Counter
def A369787(n):
    c = {0:1}
    for k in range(1,n+1):
        m, d = k*(k+1)>>1, Counter(c)
        for j in c:
            d[j+m] += c[j]
        c = d
    return len(set(c.values()))+int(max(c)+1>len(c)) # Chai Wah Wu, Feb 01 2024

A303905 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(k(k+1)/2)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 24, 24, 26, 29, 30, 31, 34, 36, 37, 41, 44, 44, 47, 50, 52, 56, 59, 62, 65, 67, 70, 73, 75, 79, 85, 89, 91, 96, 100, 102, 108, 113, 116, 123, 129, 132, 137, 142, 147, 153, 158, 162, 169, 176, 182, 190, 196, 201

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Partial sums of A024940.

Index entries for sequences related to partitions

Cf. A000217, A024940, A036469, A038348, A061208, A248801, A302835, A303668.

nmax = 69; CoefficientList[Series[1/(1 - x) Product[1 + x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (2^(1/3) * (sqrt(2) - 1)^(1/3) * sqrt(3) * Pi^(2/3) * Zeta(3/2)^(1/3) * n^(1/6)). - Vaclav Kotesovec, May 04 2018

cp's OEIS Frontend

A369787 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k*(k+1)/2)).

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A303905 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(k(k+1)/2)).

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cp's OEIS Frontend

A369787 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k*(k+1)/2)).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

A303905 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k*(k+1)/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A303905 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(k(k+1)/2)).