What's wrong with the Sum function?
I can confirm that.
Testcode: (The sums are giving out different identities (the second one is wrong) while NSum works as expected.)
a=Sum[1,{K,0,n},{M,0,n-K},{La,0,K},{Lb,0,n-K-M},{Lc,0,M}]
b=Sum[1,{La,0,n},{Lb,0,n-La},{Lc,0,n-La-Lb},{K,La,n-Lb},{M,Lc,n-K-Lb}]
TrueQ[FullSimplify[a==b]]
({a,b}/.n->#)&/@Range[1,10]
{NSum[1,{K,0,#},{M,0,#-K},{La,0,K},{Lb,0,#-K-M},{Lc,0,M}],NSum[1,{La,0,#},{Lb,0,#-La},{Lc,0,#-La-Lb},{K,La,#-Lb},{M,Lc,#-K-Lb}]}&/@Range[1,10]
Seems to be a bug in Sum
.
$Version
11.0.0 for Microsoft Windows (64-bit) (July 28, 2016)
Also confirmed by my other machine:
10.4.0 for Microsoft Windows (64-bit) (February 26, 2016)
$Version
"11.0.1 for Mac OS X x86 (64-bit) (September 21, 2016)"
Clear[f, g]
f[n_] := Sum[
1, {K, 0, n}, {M, 0, n - K}, {La, 0, K}, {Lb, 0, n - K - M}, {Lc, 0, M}]
g[n_] := Sum[
1, {La, 0, n}, {Lb, 0, n - La}, {Lc, 0, n - La - Lb}, {K, La, n - Lb}, {M,
Lc, n - K - Lb}]
The symbolic closed-form functions are not equivalent.
f[n] // FullSimplify
(* 1/120 (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) *)
g[n] // Simplify
(* 1/12 (1 + n) (2 + n)^2 (3 + n) *)
Solve[f[n] == g[n], n]
(* {{n -> -3}, {n -> -2}, {n -> -1}, {n -> 0}, {n -> 1}} *)
However, FindSequenceFunction
applied to sequences generated from either f
or g
give the symbolic closed-form function for f
f[n] == FindSequenceFunction[f /@ Range[7], n] // Simplify
(* True *)
f[n] == FindSequenceFunction[g /@ Range[7], n] // Simplify
(* True *)