A306156 Inverse Weigh transform of 2^n.
2, 3, 2, 6, 6, 11, 18, 36, 56, 105, 186, 346, 630, 1179, 2182, 4116, 7710, 14588, 27594, 52482, 99858, 190743, 364722, 699216, 1342176, 2581425, 4971008, 9587574, 18512790, 35792449, 69273666, 134219796, 260300986, 505294125, 981706806, 1908881548, 3714566310
Offset: 1
Keywords
Examples
(1+x)^2*(1+x^2)^3*(1+x^3)^2*(1+x^4)^6* ... = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + ... .
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..3000
- Christian G. Bower, PARI programs for transforms, 2007.
- N. J. A. Sloane, Maple programs for transforms, 2001-2020.
Crossrefs
Formula
Product_{k>=1} (1+x^k)^a(k) = 1/(1-2x).
a(n) = (1/n) * (2^n + Sum_{d
A038069 Product_{k>=1} ((1 + x^k)^a(k)) = 1 + 4x.
4, -6, 20, -66, 204, -670, 2340, -8226, 29120, -104754, 381300, -1398410, 5162220, -19172790, 71582716, -268439586, 1010580540, -3817733920, 14467258260, -54975633702, 209430785460, -799644629550, 3059510616420
Offset: 1
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..1000
- N. J. A. Sloane, Weigh transform
Formula
Dirichlet convolution of A038065 with characteristic function of powers of 2.
a(n) = (1/n)*(-(-4)^n + Sum_{dSeiichi Manyama, Jun 23 2018
A306157 Inverse Weigh transform of 3^n.
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
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
- Seiichi Manyama, Table of n, a(n) for n = 1..2000
- Christian G. Bower, PARI programs for transforms, 2007.
- N. J. A. Sloane, Maple programs for transforms, 2001-2020.
Crossrefs
Formula
Product_{k>=1} (1+x^k)^a(k) = 1/(1-3x).
a(n) = (1/n) * (3^n + Sum_{d
A306159 Inverse Weigh transform of 5^n.
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
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
- Seiichi Manyama, Table of n, a(n) for n = 1..1000
- Christian G. Bower, PARI programs for transforms, 2007.
- N. J. A. Sloane, Maple programs for transforms, 2001-2020.
Crossrefs
Formula
Product_{k>=1} (1+x^k)^a(k) = 1/(1-5x).
a(n) = (1/n) * (5^n + Sum_{d
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.
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
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
- Christian G. Bower, PARI programs for transforms, 2007.
- N. J. A. Sloane, Maple programs for transforms, 2001-2020.
Formula
A(n,k) = (1/n) * (k^n + Sum_{d
Product_{n>=1} (1 + x^n)^A(n,k) = 1/(1 - k*x).