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gp > for(n=1,10,print(n";",\
sum(k=0,n,binomial(n,k)*(k+1)^(k)*(n-k+1)^(n-k-1)), " VS ",(n+2)^(n)))
1;3 VS 3
2;16 VS 16
3;125 VS 125
4;1296 VS 1296
5;16807 VS 16807
6;262144 VS 262144
7;4782969 VS 4782969
8;100000000 VS 100000000
9;2357947691 VS 2357947691
10;61917364224 VS 61917364224
gp > for(n=1,10,print(n";",\
sum(k=0,n,binomial(n,k)*(k+1)^(k-1)*(n-k+1)^(n-k-1)), " VS ",2*(n+2)^(n-1)))
1;2 VS 2
2;8 VS 8
3;50 VS 50
4;432 VS 432
5;4802 VS 4802
6;65536 VS 65536
7;1062882 VS 1062882
8;20000000 VS 20000000
9;428717762 VS 428717762
10;10319560704 VS 10319560704
-------------------------------------------------------
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gp > F(n)=if(n<2,1,\
sum(i=0,n-1,binomial(n-1,i)*memorize(F,i)*(i+1)*memorize(F,n-i-1)));
gp > for(n=0,10,print(n+1";",F(n)," VS ",(n+1)^(n-1)))
1;1 VS 1
2;1 VS 1
3;3 VS 3
4;16 VS 16
5;125 VS 125
6;1296 VS 1296
7;16807 VS 16807
8;262144 VS 262144
9;4782969 VS 4782969
10;100000000 VS 100000000
11;2357947691 VS 2357947691
-------------------------------------------------------
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gp > for(n=1,10,print(n";",\
sum(k=1,n,binomial(n-1,k-1)*k^(k-2)*(n-k)^(n-k)), " VS ",n^(n-1)))
1;1 VS 1
2;2 VS 2
3;9 VS 9
4;64 VS 64
5;625 VS 625
6;7776 VS 7776
7;117649 VS 117649
8;2097152 VS 2097152
9;43046721 VS 43046721
10;1000000000 VS 1000000000
gp > for(n=1,10,print(n";",\
-sum(k=1,n+1,(-1)^k*k*binomial(n+1,k)*(n+1)^(n-k)), " VS ",n^n))
1;1 VS 1
2;4 VS 4
3;27 VS 27
4;256 VS 256
5;3125 VS 3125
6;46656 VS 46656
7;823543 VS 823543
8;16777216 VS 16777216
9;387420489 VS 387420489
10;10000000000 VS 10000000000
gp > for(n=1,10,print(n";",\
sum(k=0,n,binomial(n,k)*Derange(k+1)*(n+1)^(n-k)), " VS ",n^(n+1)))
1;1 VS 1
2;8 VS 8
3;81 VS 81
4;1024 VS 1024
5;15625 VS 15625
6;279936 VS 279936
7;5764801 VS 5764801
8;134217728 VS 134217728
9;3486784401 VS 3486784401
10;100000000000 VS 100000000000
ããã«Derange(n)ïŒ1ïœnã®æ°åã§ã®å®å
šé åã®åæ°ã瀺ãã(derangement)
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gp > for(n=1,10,print(n";",\
-sum(k=1,n+1,(-1)^k*k*binomial(n+1,k)*(n+1)^(n-k)), " VS ",n^n))
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