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 1-7 of 7 results.

A038067 Product_{k>=1} (1 + x^k)^a(k) = 1 + 2x.

Original entry on oeis.org

2, -1, 2, -4, 6, -9, 18, -34, 56, -99, 186, -344, 630, -1161, 2182, -4114, 7710, -14532, 27594, -52476, 99858, -190557, 364722, -699214, 1342176, -2580795, 4971008, -9587556, 18512790, -35790267, 69273666, -134219794, 260300986
Offset: 1

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Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A038063-A038070, A306156.

Formula

Dirichlet convolution of A038063 with characteristic function of powers of 2.
a(n) = (1/n) * (-(-2)^n + Sum_{dVladeta Jovovic, Sep 06 2002

A306157 Inverse Weigh transform of 3^n.

Original entry on oeis.org

3, 6, 8, 24, 48, 124, 312, 834, 2184, 5928, 16104, 44344, 122640, 341796, 956576, 2690844, 7596480, 21524412, 61171656, 174342192, 498111952, 1426419852, 4093181688, 11767919284, 33891544368, 97764131640, 282429535752, 817028472936, 2366564736720, 6863038212784
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2018

Keywords

Examples

			(1+x)^3*(1+x^2)^6*(1+x^3)^8*(1+x^4)^24* ... = 1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + ... .

Links

Crossrefs

Inverse Weigh transform of b^n: A306156 (b=2), this sequence (b=3), A306158 (b=4), A306159 (b=5).
Cf. A000244, A038068.

Formula

Product_{k>=1} (1+x^k)^a(k) = 1/(1-3x).
a(n) = (1/n) * (3^n + Sum_{d

A306158 Inverse Weigh transform of 4^n.

Original entry on oeis.org

4, 10, 20, 70, 204, 690, 2340, 8230, 29120, 104958, 381300, 1398430, 5162220, 19175130, 71582716, 268439590, 1010580540, 3817763040, 14467258260, 54975633906, 209430785460, 799645010850, 3059510616420, 11728124726270, 45035996273664, 173215372864590
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2018

Keywords

Examples

			(1+x)^4*(1+x^2)^10*(1+x^3)^20*(1+x^4)^70* ... = 1 + 4*x + 16*x^2 + 64*x^3 + 256*x^4 + ... .

Links

Crossrefs

Inverse Weigh transform of b^n: A306156 (b=2), A306157 (b=3), this sequence (b=4), A306159 (b=5).
Cf. A000302, A038069.

Formula

Product_{k>=1} (1+x^k)^a(k) = 1/(1-4x).
a(n) = (1/n) * (4^n + Sum_{d

A306159 Inverse Weigh transform of 5^n.

Original entry on oeis.org

5, 15, 40, 165, 624, 2620, 11160, 48915, 217000, 976872, 4438920, 20346320, 93900240, 435970980, 2034504992, 9536767665, 44878791360, 211927733500, 1003867701480, 4768372070592, 22706531339280, 108372083629260, 518301258916440, 2483526875798820
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2018

Keywords

Examples

			(1+x)^5*(1+x^2)^15*(1+x^3)^40*(1+x^4)^165* ... = 1 + 5*x + 25*x^2 + 125*x^3 + 625*x^4 + ... .

Links

Crossrefs

Inverse Weigh transform of b^n: A306156 (b=2), A306157 (b=3), A306158 (b=4), this sequence (b=5).
Cf. A000351, A038070.

Formula

Product_{k>=1} (1+x^k)^a(k) = 1/(1-5x).
a(n) = (1/n) * (5^n + Sum_{d

A383034 Inverse Weigh transform of 2^(n-1).

Original entry on oeis.org

1, 2, 2, 5, 6, 11, 18, 35, 56, 105, 186, 346, 630, 1179, 2182, 4115, 7710, 14588, 27594, 52482, 99858, 190743, 364722, 699216, 1342176, 2581425, 4971008, 9587574, 18512790, 35792449, 69273666, 134219795, 260300986, 505294125, 981706806, 1908881548, 3714566310
Offset: 1

Views

Author

Seiichi Manyama, Apr 13 2025

Keywords

Links

Crossrefs

Column k=2 of A383033.
Cf. A000079, A209229, A306156.

Formula

a(n) = (1/n) * (2^n - 1 + Sum_{d
a(n) = A306156(n) - A209229(n).
Product_{k>=1} (1 + x^k)^a(k) = (1 - x)/(1 - 2*x).

A383035 Inverse Weigh transform of 3^(n-1).

Original entry on oeis.org

1, 3, 6, 18, 42, 113, 294, 798, 2128, 5823, 15918, 43998, 122010, 340617, 954394, 2686728, 7588770, 21509824, 61144062, 174289710, 498012094, 1426229109, 4092816966, 11767220068, 33890202192, 97761550215, 282424564744, 817018885362, 2366546223930, 6863002420335
Offset: 1

Views

Author

Seiichi Manyama, Apr 13 2025

Keywords

Links

Crossrefs

Column k=3 of A383033.
Cf. A306156, A306157.

Formula

a(n) = (1/n) * (3^n - 2^n + Sum_{d
a(n) = A306157(n) - A306156(n).
Product_{k>=1} (1 + x^k)^a(k) = (1 - 2*x)/(1 - 3*x).

A383023 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^j.

Original entry on oeis.org

1, 2, 1, 3, 3, 0, 4, 6, 2, 1, 5, 10, 8, 6, 0, 6, 15, 20, 24, 6, 0, 7, 21, 40, 70, 48, 11, 0, 8, 28, 70, 165, 204, 124, 18, 1, 9, 36, 112, 336, 624, 690, 312, 36, 0, 10, 45, 168, 616, 1554, 2620, 2340, 834, 56, 0, 11, 55, 240, 1044, 3360, 7805, 11160, 8230, 2184, 105, 0
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Examples

			Square array begins:
  1,  2,   3,    4,     5,     6,      7, ...
  1,  3,   6,   10,    15,    21,     28, ...
  0,  2,   8,   20,    40,    70,    112, ...
  1,  6,  24,   70,   165,   336,    616, ...
  0,  6,  48,  204,   624,  1554,   3360, ...
  0, 11, 124,  690,  2620,  7805,  19656, ...
  0, 18, 312, 2340, 11160, 39990, 117648, ...

Links

Crossrefs

Columns k=1..5 give A209229, A306156, A306157, A306158, A306159.
Cf. A074650.

Formula

A(n,k) = (1/n) * (k^n + Sum_{d
Product_{n>=1} (1 + x^n)^A(n,k) = 1/(1 - k*x).
Showing 1-7 of 7 results.

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