MOON2 - Moon Safari (Hard)

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Air is a music duo from France.
You will be told the secret of the critically acclaimed album Moon Safari: mathematics.
The goal of your new task is to compute an ethereal sum.

AirSum

Three trips on the moon are provided, Moon (easy), Moon1 (medium), Moon2 (hard) with different constraints.

Input

The first line contains an integer T, the number of test cases.
On the next T lines, you will be given three integers N, a and r.

Output

Output T lines, one for each test case, with SN,a,r = sum( a^i i^r, for i in [1..N] ).
Since the answer can get very big, output it modulo 109+7.

Example

Input:
2
3 4 5
6 7 8

Output:
16068
329990641

Explanation

The first case is, with N=3, a=4, r=5, about the sum : 4^1 × 1^5 + 4^2 × 2^5 + 4^3 × 3^5 = 4 + 512 + 15552 = 16068.
The second case is, with N=6, a=7, r=8, about the sum : 7^1 × 1^8 + 7^2 × 2^8 + 7^3 × 3^8 + 7^4 × 4^8 + 7^5 × 5^8 + 7^6 × 6^8 + 7^7 × 7^8 = 204329992069 ≡ 329990641 (mod 10^9+7).

Constraints

0 < T×r < 2 560 000
1 < N < 10^9
1 < a < 10^9

Uniform random input in the range.
More precise information : there are 5 input files.
In0 : T<20000 and r <= 128
In1 : T<2000 and r <= 1250
In2 : T<200 and r <= 11000
In3 : T<20 and r <= 100000
In4 : T<2 and r <= 1000000

Information

This trip can be done with a O(T×r×log(N)) method and some interpreted languages.
Edited(2017-02-11, after compiler changes).
My non optimal Py3 code got AC under 5s for each of the 5 input files.
@speed addicts : My fastest C code got AC in 0.42s. (approx 0.08s per file)
Good luck and have fun ;-)


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pranav_arora: 2015-07-21 04:00:05

Can you please give some hints? I have been trying this problem from days but cannot improve the recurrence I set up for the medium version. :(

Please help. Thanks!

=(Francky)=> A new method is need ; check your complexity ; good luck.

Last edit: 2015-07-21 09:35:10
Min_25: 2014-10-24 12:18:37

Finally done. :)
--ans(Francky)--> Well done.

Last edit: 2014-10-24 12:43:27
Francky: 2014-06-12 22:18:03

Congratulations to Robert Gerbicz as the first solver. Well done.
(gerrob) Thanks!

Last edit: 2014-06-14 18:29:02

Added by:Francky
Date:2014-06-07
Time limit:9s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ASM64
Resource:Own problem