A158465 -id:A158465 - cp's OEIS frontend

A369358 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 15, 25, 59, 109, 227, 525, 1321, 2917, 7085, 15893, 38759, 90179, 223933, 534867, 1339691, 3246961, 8296441, 20426971, 52715563, 131480623, 342491253, 864759837, 2270860455, 5793103989, 15316065497, 39429185075, 105008858223

Offset: 0

Ilya Gutkovskiy, Jan 21 2024

nonn

Cf. A000583, A007576, A158465, A350249.

Table[Coefficient[Product[x^(k^4) + 1 + 1/x^(k^4), {k, 1, n}], x, 0], {n, 0, 25}]

a(26)-a(37) from Alois P. Heinz, Jan 21 2024

A369629 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 0 or 1.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 16, 18, 0, 21, 32, 100, 126, 0, 424, 510, 0, 1428, 2792, 5988, 9786, 7, 29058, 45106, 22, 150437, 276828, 473854, 836737, 1838, 2455340, 4777436, 15847, 14696425, 27466324, 46429640, 83010230, 738627

Offset: 0

Ilya Gutkovskiy, Jan 28 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..73

Cf. A000583, A025591, A111253, A158465, A350403, A350861.

from itertools import count, islice
from collections import Counter
def A369629_gen(): # generator of terms
    ccount = Counter({0:1})
    yield 1
    for i in count(1):
        bcount = Counter()
        for a in ccount:
            bcount[a+(j:=i**4)] += ccount[a]
            bcount[a-j] += ccount[a]
        ccount = bcount
        yield(ccount[0]+ccount[1])
A369629_list = list(islice(A369629_gen(),20)) # Chai Wah Wu, Jan 29 2024

a(46)-a(50) from Chai Wah Wu, Jan 29 2024

A369732 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 21, 0, 0, 126, 0, 0, 0, 0, 1428, 0, 0, 9786, 7, 0, 0, 22, 150437, 0, 0, 836737, 1838, 0, 0, 15847, 14696425, 0, 0, 83010230, 738627, 0, 0, 6024822, 1640652994, 0, 0, 10271377082, 226033104, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A000583, A063866, A111253, A158465, A368206, A369629, A369730, A369731.

a(n) = [x^1] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

a(46)-a(60) from Alois P. Heinz, Jan 30 2024

A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 8, 13, 0, 0, 272, 0, 0, 0, 5400, 8915, 0, 0, 30433, 1590, 0, 0, 68638, 73470, 0, 0, 90808, 6638072, 0, 0, 127356, 319803, 0, 0, 20130146, 559282596, 0, 0, 1507066936, 3820244957, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A063890, A158465, A348165, A348892.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n, n)):
seq(a(n), n=0..43);  # Alois P. Heinz, Jan 25 2024

a(46)-a(59) from Alois P. Heinz, Jan 25 2024

A368348 a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 19, 26, 0, 0, 40, 129, 0, 0, 616, 785, 0, 0, 4080, 9309, 0, 0, 44775, 72659, 0, 0, 430297, 781505, 0, 0, 3934457, 7765047, 0, 0, 44740433, 78818429, 0, 0, 463089552, 900950811, 0, 0, 5344766190, 9806206864, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A063890, A158465, A368243, A368845.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^4, n)):
seq(a(n), n=0..43);  # Alois P. Heinz, Jan 25 2024

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30];
a[n_] := If[Mod[n, 4] > 1, 0, b[n^4, n]];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Feb 14 2025, after Alois P. Heinz *)

a(46)-a(59) from Alois P. Heinz, Jan 25 2024

cp's OEIS Frontend

A369358 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

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A369629 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 0 or 1.

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A369732 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 1.

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A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

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A368348 a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

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Maple

Mathematica

Extensions