cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A321435 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, 2, 0, 1, 2, 0, 3, 0, 2, 3, 1, 4, 1, 2, 4, 1, 6, 3, 4, 6, 2, 7, 5, 6, 8, 5, 9, 7, 9, 10, 9, 12, 10, 13, 14, 13, 18, 13, 19, 17, 18, 25, 19, 28, 24, 25, 33, 26, 36, 35, 33, 46, 35, 47, 48, 44, 61, 48, 62, 65, 60, 78, 68, 79, 87, 79, 101, 93
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2018

Keywords

Links

Crossrefs

Cf. A025426, A321423, A321428, A321436, A321437.

Programs

  • Maple
    N:= 100: # for a(0)..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

Formula

G.f.: Product_{k>0} (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

Views

Author

Seiichi Manyama, Nov 09 2018

Keywords

Links

Crossrefs

Cf. A025426, A321430, A321435, A321437.

Programs

  • Maple
    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

Formula

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

A321431 Expansion of Product_{i>0, j>0} 1/(1 - x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 2, 2, 2, 7, 2, 7, 6, 7, 14, 8, 16, 18, 16, 34, 20, 38, 40, 39, 68, 54, 78, 91, 84, 143, 116, 161, 184, 185, 270, 252, 312, 372, 372, 518, 494, 607, 704, 736, 944, 965, 1130, 1311, 1378, 1723, 1784, 2081, 2360, 2548, 3048, 3250, 3704, 4196, 4544
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2018

Keywords

Links

Crossrefs

Convolution inverse of A321430.
Cf. A063725, A321428, A321433, A321437.

Programs

  • Mathematica
    nmax = 100; A063725 = Rest[CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^2/4, {x, 0, nmax}], x]]; s = 1; Do[s *= Sum[(-1)^j*Binomial[A063725[[k]], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2018 *)

Formula

G.f.: Product_{k>0} 1/(1 - x^k)^A063725(k).

A321445 Expansion of Product_{0 < i < j} 1/(1 - x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 1, 0, 4, 0, 1, 2, 0, 5, 2, 2, 2, 1, 8, 2, 2, 4, 3, 9, 4, 5, 5, 6, 14, 5, 7, 9, 8, 17, 10, 12, 10, 12, 26, 15, 18, 17, 19, 32, 22, 26, 23, 28, 47, 29, 36, 38, 39, 62, 44, 52, 52, 54, 85, 63, 70, 75, 76, 111, 90, 98, 106
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2018

Keywords

Crossrefs

Cf. A025441, A319734, A321437, A321444.

Formula

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

A321459 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 5, 0, 2, 6, 0, 4, 10, 1, 6, 13, 3, 10, 19, 5, 15, 26, 10, 23, 37, 16, 32, 52, 26, 48, 74, 38, 70, 100, 59, 101, 138, 85, 144, 189, 126, 204, 256, 180, 284, 345, 262, 392, 468, 372, 537, 629, 522, 730, 847, 723, 992, 1134, 998, 1336
Offset: 0

Views

Author

Seiichi Manyama, Nov 10 2018

Keywords

Crossrefs

Cf. A025427, A321433, A321437, A321457, A321458.

Formula

G.f.: Product_{k>0} 1/(1 - x^k)^A025427(k).
Showing 1-5 of 5 results.

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