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 1-8 of 8 results.

A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

Original entry on oeis.org

1, 2, 4, 8, 16, 40, 88, 222, 570, 1564, 4516, 13874, 41866, 137432, 442964, 1492610, 4998674, 17204844, 59175316, 207299554, 727137516, 2582078416, 9179001124, 32943918428, 118453240846, 428937325964, 1556421977612, 5676923326262, 20754245720206
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2022

Keywords

Links

Crossrefs

Cf. A000980, A047653, A158092, A350249, A350881.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1/x^(n^2))^2*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 28 2022
  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Formula

Conjecture: a(n) ~ sqrt(5) * 4^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Feb 05 2022

A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 11, 27, 61, 133, 311, 761, 1839, 4575, 11573, 29641, 76487, 199617, 524067, 1384697, 3681069, 9841217, 26437741, 71369101, 193496241, 526685793, 1438816755, 3944034221, 10845006963, 29908325821, 82707648985, 229306378067, 637283978821
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Comments

All terms are odd.
a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * i*(i+1)/2 with c_i in {-1,0,1}.

Links

Crossrefs

Cf. A000217, A007576, A158380, A350249.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i*(i+1)*(i+2)/6, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+i*(i+1)/2, i-1)+b(abs(n-i*(i+1)/2), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024
  • Mathematica
    Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, 0], {n, 0, 31}]

Formula

a(n) ~ sqrt(5) * 3^(n + 1/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jan 22 2024

A369345 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 5, 17, 31, 61, 139, 309, 701, 1651, 3849, 8929, 22295, 53777, 131025, 335619, 837999, 2107947, 5484373, 14071891, 36275323, 95881995, 250956301, 659257445, 1763642977, 4685724391, 12496708267, 33766814039, 90846586161, 245197523769
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Comments

All terms are odd.

Crossrefs

Cf. A000578, A007576, A158118, A350249.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,
     `if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 21 2024
  • Mathematica
    Table[Coefficient[Product[x^(k^3) + 1 + 1/x^(k^3), {k, 1, n}], x, 0], {n, 0, 33}]

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 21 2024

Keywords

Crossrefs

Cf. A000583, A007576, A158465, A350249.

Programs

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

Extensions

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

A369433 a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

Original entry on oeis.org

1, 1, 0, 1, 3, 5, 7, 13, 35, 82, 168, 409, 1035, 2540, 6262, 16068, 41474, 107259, 279256, 736359, 1953946, 5205746, 13938670, 37567522, 101675407, 276158642, 752927255, 2060852216, 5658658210, 15582628517, 43032891276, 119166025289, 330808837377
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2024

Keywords

Crossrefs

Cf. A000290, A316706, A348165, A350249.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, n], {n, 0, 32}]

A369434 a(n) = [x^(n^2)] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 13, 31, 75, 155, 344, 808, 2019, 5136, 13422, 34720, 91055, 238447, 630753, 1678780, 4502862, 12135507, 32873145, 89324119, 243745113, 667153916, 1832553339, 5048767393, 13950607375, 38649239592, 107345311219, 298820158401, 833680894927
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2024

Keywords

Crossrefs

Cf. A000290, A316706, A350249, A368243.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, n^2], {n, 0, 33}]

A369734 Number of solutions to 1^2*k_1 + 2^2*k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 8, 17, 35, 79, 177, 409, 995, 2475, 6336, 16078, 41401, 107304, 279550, 736032, 1950000, 5199850, 13950852, 37576658, 101670863, 276228026, 753114256, 2060995699, 5658663542, 15583477334, 43039552072, 119179072495, 330836308272, 920537185436
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A350249, A369433, A369628, A369735.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 32}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

A369389 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2))^2.

Original entry on oeis.org

1, 3, 9, 35, 141, 745, 3955, 23985, 155527, 1060941, 7393765, 53041015, 387815175, 2882682967, 21715452927, 165583974835, 1275674593889, 9918184576835, 77738274996385, 613753581566079, 4877383708962749, 38989308129231703, 313354624116918229, 2530796548734844153
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2024

Keywords

Crossrefs

Cf. A047653, A156181, A350249, A350495.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 23}]
Showing 1-8 of 8 results.

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