A321435 -id:A321435 - cp's OEIS frontend

A321437 Expansion of Product_{1 <= i <= j} 1/(1 - x^(i^2 + j^2)).

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 1, 4, 3, 4, 5, 5, 6, 9, 6, 13, 8, 14, 13, 15, 19, 21, 21, 29, 24, 38, 32, 42, 44, 51, 56, 65, 65, 83, 79, 102, 99, 120, 125, 144, 154, 176, 183, 213, 219, 262, 266, 311, 322, 369, 392, 437, 465, 526, 550, 635, 650, 747, 781, 871, 937

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321436.

Cf. A025426, A321431, A321435.

G.f.: Product_{k>0} 1/(1 - x^k)^A025426(k).

A321436 Expansion of Product_{1 <= i <= j} (1 - x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, -1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 0, 1, 0, -2, 1, 1, -2, 1, 2, 0, -1, 0, -1, 0, 2, -2, 1, 1, -4, 0, 3, -1, -1, 3, -2, -1, 0, -4, 5, 2, -3, 2, 3, -5, -3, 6, -3, -1, 0, -2, 1, 1, 0, 2, 7, -7, 0, 7, -9, -2, 4, -3, 2, 6, -9, 2, 12, -12, 1, 9, -11, -3, 7, 0, -1, 5

Offset: 0

Seiichi Manyama, Nov 09 2018

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025426, A321430, A321435, A321437.

N:= 100: # for a(1)..a(N)
P:= 1:
for i from 1 to floor(sqrt(N)) do
  for j from i while i^2 + j^2 <= N do
    P:= P * (1 - x^(i^2 + j^2))
od od:
S:= series(P,x,N+1):
seq(coeff(S,x,k),k=0..N): # Robert Israel, Apr 21 2024

G.f.: Product_{k>0} (1 - x^k)^A025426(k).

A319734 Expansion of Product_{0 < i < j} (1 + x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 2, 4, 3, 3, 4, 3, 5, 3, 3, 6, 5, 6, 4, 6, 8, 6, 6, 7, 8, 9, 7, 8, 10, 10, 12, 10, 12, 13, 11, 15, 15, 14, 15, 14, 20, 19, 16, 21, 20, 25, 22, 22, 27

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Cf. A025441, A321435, A320332, A321444, A321445.

G.f.: Product_{k>0} (1 + x^k)^A025441(k).

A321457 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 3, 3, 1, 4, 4, 2, 5, 4, 3, 7, 7, 5, 9, 10, 7, 11, 14, 9, 15, 19, 12, 22, 23, 17, 30, 29, 23, 41, 37, 32, 54, 46, 45, 68, 59, 63, 85, 79, 85, 107, 103, 108, 136, 136, 139, 174, 177, 178, 222, 225, 226, 287, 282, 290

Offset: 0

Seiichi Manyama, Nov 10 2018

nonn

Cf. A025427, A321429, A321435, A321458, A321459.

G.f.: Product_{k>0} (1 + x^k)^A025427(k).

cp's OEIS Frontend

A321437 Expansion of Product_{1 <= i <= j} 1/(1 - x^(i^2 + j^2)).

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A321436 Expansion of Product_{1 <= i <= j} (1 - x^(i^2 + j^2)).

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A319734 Expansion of Product_{0 < i < j} (1 + x^(i^2 + j^2)).

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A321457 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

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