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Closed Closure (Posted on 2011-04-28) Difficulty: 3 of 5
Determine, with proof, the closed form of following expression in terms of n

Σj=1 to n  j2*(j+1)*(nCj)

See The Solution Submitted by K Sengupta    
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Some Thoughts here are some numbers in case they help | Comment 1 of 3

Some values are listed below, in case that helps:

  5   for N=1 to 40
  6    Tot=0
 10    for J=1 to N
 20     Term=J*J*(J+1)*combi(N,J)
 30     Tot=Tot+Term
 40    next J
 45    print N,Tot
 50   next N
 
n       sum
1       2
2       16
3       78
4       304
5       1040
6       3264
7       9632
8       27136
9       73728
10      194560
11      501248
12      1265664
13      3141632
14      7684096
15      18554880
16      44302336
17      104726528
18      245366784
19      570294272
20      1315962880
21      3016753152
22      6874464256
23      15579742208
24      35131490304
25      78852915200
26      176227876864
27      392284864512
28      869999312896
29      1922803171328
30      4235911495680
31      9303436034048
32      20375324852224
33      44504451121152
34      96963181674496
35      210754045214720
36      457053239771136
37      989079428661248
38      2136076214861824
39      4604342380265472
40      9906599766261760
The terms are broken down below:
 5   for N=1 to 9
 6    Tot=0:print N,
10    for J=1 to N
20     Term=J*J*(J+1)*combi(N,J):print Term;
30     Tot=Tot+Term
40    next J
45    print ":";Tot
50   next N
  
n       terms and total
1       2 : 2
2       4  12 : 16
3       6  36  36 : 78
4       8  72  144  80 : 304
5       10  120  360  400  150 : 1040
6       12  180  720  1200  900  252 : 3264
7       14  252  1260  2800  3150  1764  392 : 9632
8       16  336  2016  5600  8400  7056  3136  576 : 27136
9       18  432  3024  10080  18900  21168  14112  5184  810 : 73728

  Posted by Charlie on 2011-04-28 11:44:18
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