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
A038068 Product_{k>=1}(1 + x^k)^a(k) = 1 + 3x.
3, -3, 8, -21, 48, -116, 312, -831, 2184, -5880, 16104, -44336, 122640, -341484, 956576, -2690841, 7596480, -21522228, 61171656, -174342144, 498111952, -1426403748, 4093181688, -11767919276, 33891544368, -97764009000
Offset: 1
Keywords
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.
Formula
Dirichlet convolution of A038064 with characteristic function of powers of 2.
a(n) = (1/n) * (-(-3)^n + Sum_{dSeiichi Manyama, Jun 23 2018
A306158 Inverse Weigh transform of 4^n.
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
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
- 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/(1-4x).
a(n) = (1/n) * (4^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
A383035 Inverse Weigh transform of 3^(n-1).
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
Keywords
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.
Formula
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).