# Light OJ: 1132 – Summing up Powers

Problem Link : http://lightoj.com:81/volume/problem/1132

Soluton Idea:
————–

f(x)=1^k+2^k+3^k+……….+x^k
f(1)=1
f(x+1)=f(x)+(x+1)^k

from Binomial Co efficient we know –
(x+1)^n= nC0 x^n + nC1 x^n-1 + nC2 x^n-2+…………..+ nCn-1 x^1 + nCn x^0

Now make a matrix with co efficient of x so that we can find f(x+1) from f(x).

details: http://lbv-pc.blogspot.com/2012/05/summing-up-powers.html

```

#include &lt;bits/stdc++.h&gt;

#define pii              pair &lt;int,int&gt;
#define pll              pair &lt;long long,long long&gt;
#define sc               scanf
#define pf               printf
#define Pi               2*acos(0.0)
#define ms(a,b)          memset(a, b, sizeof(a))
#define pb(a)            push_back(a)
#define MP               make_pair
#define db               double
#define ll               long long
#define EPS              10E-10
#define ff               first
#define ss               second
#define sqr(x)           (x)*(x)
#define D(x)             cout&lt;&lt;#x &quot; = &quot;&lt;&lt;(x)&lt;&lt;endl
#define VI               vector &lt;int&gt;
#define DBG              pf(&quot;Hi\n&quot;)
//#define MOD              1000000007
#define CIN              ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0)
#define SZ(a)            (int)a.size()
#define sf(a)            scanf(&quot;%d&quot;,&amp;a)
#define sfl(a)           scanf(&quot;%lld&quot;,&amp;a)
#define sff(a,b)         scanf(&quot;%d %d&quot;,&amp;a,&amp;b)
#define sffl(a,b)        scanf(&quot;%lld %lld&quot;,&amp;a,&amp;b)
#define sfff(a,b,c)      scanf(&quot;%d %d %d&quot;,&amp;a,&amp;b,&amp;c)
#define sfffl(a,b,c)     scanf(&quot;%lld %lld %lld&quot;,&amp;a,&amp;b,&amp;c)
#define stlloop(v)       for(__typeof(v.begin()) it=v.begin();it!=v.end();it++)
#define loop(i,n)        for(int i=0;i&lt;n;i++)
#define loop1(i,n)       for(int i=1;i&lt;=n;i++)
#define REP(i,a,b)       for(int i=a;i&lt;b;i++)
#define RREP(i,a,b)      for(int i=a;i&gt;=b;i--)
#define TEST_CASE(t)     for(int z=1;z&lt;=t;z++)
#define PRINT_CASE       printf(&quot;Case %d: &quot;,z)
#define CASE_PRINT       cout&lt;&lt;&quot;Case &quot;&lt;&lt;z&lt;&lt;&quot;: &quot;
#define all(a)           a.begin(),a.end()
#define intlim           2147483648
#define infinity         (1&lt;&lt;28)
#define ull              unsigned long long
#define gcd(a, b)        __gcd(a, b)
#define lcm(a, b)        ((a)*((b)/gcd(a,b)))

using namespace std;

/*----------------------Graph Moves----------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*------------------------------------------------*/

//int Set(int N,int pos){return N=N | (1&lt;&lt;pos);}
//int reset(int N,int pos){return N= N &amp; ~(1&lt;&lt;pos);}
//bool check(int N,int pos){return (bool)(N &amp; (1&lt;&lt;pos));}
/*------------------------------------------------*/

/*----------------------Matrix-----------------------*/

// int MOD=
// ll MOD= (1LL)&lt;&lt;32;

struct matrix
{
unsigned int mat[52][52];
int row,col;

matrix()
{
memset(mat,0,sizeof mat);
}
matrix(int a, int b)
{
row=a,col=b;
memset(mat,0,sizeof mat);
}

matrix operator*(const matrix &amp;p) const
{
assert(col == p.row);
matrix temp;
temp.row = row;
temp.col = p.col;
for (int i = 0; i &lt; temp.row; i++)
{
for (int j = 0; j &lt; temp.col; j++)
{
ll sum = 0;
for (int k = 0; k &lt;col;  k++)
{
sum += ((mat[i][k]) * (p.mat[k][j]));
//                    sum%=MOD;
}
temp.mat[i][j] = sum;
}
}
return temp;
}
matrix operator+ (const matrix &amp;p) const
{
assert(row==p.row &amp;&amp; col==p.col);
matrix temp;
temp.row=row;
temp.col=col;
for(int i=0;i&lt;temp.row;i++)
{
for(int j=0;j&lt;temp.col;j++)
temp.mat[i][j]=((mat[i][j])+(p.mat[i][j]));;
}
return temp;
}

matrix identity()
{
matrix temp;
temp.row=row;
temp.col=col;
for(int i=0;i&lt;row;i++)
temp.mat[i][i]=1;
return temp;
}

matrix pow(ll pow)
{
matrix temp=(*this);
matrix ret=(*this).identity();
while(pow)
{
if(pow % 2==1)
ret=ret*temp;
temp=temp*temp;
pow/=2;
}
return ret;
}

void show()
{
printf(&quot;-----------------------------\n&quot;);
for(int i=0;i&lt;row;i++)
{
for(int j=0;j&lt;col;j++)
printf(&quot;%lld &quot;,mat[i][j]);
printf(&quot;\n&quot;);
}
printf(&quot;-----------------------------\n&quot;);
}

};

/*--------------------------Matrix End---------------------*/

unsigned int nCr[55][55];

void compute_nCr()
{
nCr[0][0]=nCr[1][1]=1;
for(int i=1;i&lt;52;i++)
{
nCr[i][0]=1;
for(int j=1;j&lt;52;j++)
nCr[i][j]=nCr[i-1][j]+nCr[i-1][j-1];
}
}

int main()
{

//    freopen(&quot;in.txt&quot;,&quot;r&quot;,stdin);
///freopen(&quot;out.txt&quot;,&quot;w&quot;,stdout);

int t;
sf(t);
compute_nCr();

//    int a=5;
//    while(a--)
//    {
//        int a,b;
//        cin&gt;&gt;a&gt;&gt;b;
//        cout&lt;&lt;nCr[a][b]&lt;&lt;endl;
//    }

TEST_CASE(t)
{
ll n,k;
sffl(n,k);
matrix base(k+2,k+2);

base.mat[0][0]=1;

for(int i=1;i&lt;=k+1;i++)
{
base.mat[0][i]=nCr[k][i-1];
base.mat[1][i]=nCr[k][i-1];
}
for(int i=2;i&lt;=k+1;i++)
{
for(int j=i;j&lt;=k+1;j++)
base.mat[i][j]=nCr[k-i+1][j-i];
}

//        base.show();

matrix temp(k+2,1);
for(int i=0;i&lt;=k+1;i++)
temp.mat[i][0]=1;

base=base.pow(n-1);
//        base.show();
//        temp.show();
base=base*temp;
PRINT_CASE;
printf(&quot;%u\n&quot;,base.mat[0][0]);

}

return 0;
}

```
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