A039822 -id:A039822 - cp's OEIS frontend

A369790 Number of different coefficient values in expansion of Product_{k=1..n} (1-x^k)^3.

1, 4, 5, 15, 13, 31, 26, 57, 42, 91, 66, 139, 95, 209, 129, 283, 171, 365, 216, 463, 272, 573, 333, 697, 401, 825, 468, 993, 545, 1139, 629, 1315, 725, 1509, 815, 1689, 920, 1921, 1030, 2139, 1147, 2367, 1261, 2619, 1391, 2861, 1521, 3135, 1659, 3409, 1802, 3703, 1952

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A010816, A039822.

a(n) = #Set(Vec(prod(k=1, n, (1-x^k)^3)));

from collections import Counter
def A369790(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            d[j+k] -= 3*a
            d[j+2*k] += 3*a
            d[j+3*k] -= a
        c = d
    return len(set(c.values()))+int(max(c)+1>len(c)) # Chai Wah Wu, Feb 01 2024

A039829 Number of different coefficient values in expansion of Product_{i=1..n} (1 + (-q)^i).

Original entry on oeis.org

2, 2, 3, 4, 6, 10, 13, 17, 36, 48, 33, 39, 86, 100, 60, 68, 148, 166, 95, 105, 226, 248, 138, 150, 320, 346, 189, 203, 430, 460, 248, 264, 556, 590, 315, 333, 698, 736, 390, 410, 856, 898, 473, 495, 1030, 1076, 564, 588, 1220, 1270

Offset: 1

Olivier Gérard

nonn

Cf. A039822, A039828.

a(n) = #vecsort(Vec(prod(i = 1, n, (1 + (-q)^i))), , 8) \\ David A. Corneth, Aug 29 2018

A369786 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k^2)).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 6, 7, 10, 15, 21, 33, 50, 83, 126, 208, 321, 498, 688, 934, 1208, 1496, 1798, 2140, 2482, 2862, 3268, 3690, 4145, 4619, 5142, 5687, 6265, 6880, 7530, 8214, 8937, 9700, 10489, 11339, 12218, 13142, 14112, 15123, 16181, 17288, 18438, 19639, 20888

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A033461, A039822.

p = 1; Join[{1}, Table[p = Expand[p*(1 + x^(n^2))]; Length[Union[CoefficientList[p, x]]], {n, 1, 50}]] (* or *)
nmax = 50; poly = ConstantArray[0, nmax*(nmax + 1)*(2*nmax + 1)/6 + 1]; poly[[1]] = 1; poly[[2]] = 1; Flatten[{{1, 1}, Table[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, k*(k + 1)*(2*k + 1)/6, k^2, -1}]; Length[Union[poly]], {k, 2, nmax}]}] (* Vaclav Kotesovec, Feb 01 2024 *)

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

from collections import Counter
def A369786(n):
    c = {0:1}
    for k in range(1,n+1):
        m, d = k**2, 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

Conjecture: a(n) ~ n^3/6. - Vaclav Kotesovec, Feb 02 2024

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

Original entry on oeis.org

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

A369879 Number of different coefficient values in expansion of Product_{k=1..n} (1-x^k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 9, 9, 11, 12, 15, 16, 17, 21, 23, 25, 33, 35, 41, 43, 53, 55, 77, 74, 97, 85, 135, 111, 175, 131, 223, 157, 269, 187, 315, 219, 375, 245, 437, 280, 505, 320, 593, 357, 671, 398, 761, 445, 825, 489, 933, 529, 1039, 576, 1119, 627, 1211, 682, 1309, 730, 1415

Offset: 0

Seiichi Manyama, Feb 04 2024

nonn

Cf. A010815, A039822, A369790.

PARI
```
a(n) = #Set(Vec(prod(k=1, n, 1-x^k)));
```

from collections import Counter
def A369879(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            d[j+k] -= c[j]
        c = d
    return len(set(c.values()))+int(max(c)+1>len(c)) # Chai Wah Wu, Feb 04 2024

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A369790 Number of different coefficient values in expansion of Product_{k=1..n} (1-x^k)^3.

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A039829 Number of different coefficient values in expansion of Product_{i=1..n} (1 + (-q)^i).

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A369786 Number of different coefficient values in expansion of Product_{k=1..n} (1+x^(k^2)).

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

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A369879 Number of different coefficient values in expansion of Product_{k=1..n} (1-x^k).

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Python