# Is this the correct use of nested sums & binomials?

Discussion in 'Algebra' started by aidan, Oct 18, 2012.

1. ### aidan

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from The On-Line Encyclopedia of Integer Sequences!
I found a sequence that was defined by a formula:

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using OEIS
for my question, non pertinent informatiion omitted.

(Greetings from The On-Line Encyclopedia of Integer Sequences!)

Search: seq:3,1,3,7,11,21,39,71,131

Displaying 1-1 of 1 result found. page 1

A001644 a(n) = a(n-1) + a(n-2) + a(n-3), a(0)=3, a(1)=1, a(2)=3.

3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499, 2757, 5071, 9327, 17155, 31553, 58035, 106743, 196331, 361109, 664183, 1221623, 2246915, 4132721, 7601259, 13980895, 25714875, 47297029, 86992799, 160004703, 294294531, 541292033, 995591267, 1831177831

FORMULA

a(n)=n*sum(k=1..n, sum(j=n-3*k..k, binomial(j,n-3*k+2*j)*binomial(k,j))/k),
n>0, a(0)=3. [From Vladimir Kruchinin, Feb 24 2011]

f[n_] := n*Sum[ Sum[ Binomial[j, n - 3*k + 2*j]*Binomial[k, j], {j, n - 3*k, k}]/k, {k, n}]; f = 3;

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My question is:
How is this formula interpreted?

a(n) = n*sum(k=1..n, sum(j=n-3*k..k, binomial(j,n-3*k+2*j)*binomial(k,j))/k)
n>0, a(0)=3.

Binomial( n , k) ; n CHOOSE k (from n items how many ways can you choose k items)
This appears to be the meaning of the function "binomial"
which expanded results in: n!/( k! (n-k)!)
the factorial of n divided by the quantity of ( factorial of k times the factorial of (n minus k) )

This appears to be the problem
sum(j=n-3*k..k, <-- for j = n-3*k to k [ for j in the range n-3*k ... k]

and this
sum(k=1..n <-- for k = 1 to n

-------------------------------
This is a BASIC program to compute the values
n = 11
v = 0
for k = 1 to n
for j = n-3*k to k
v = v + ( binomial(j,n-3*k+2*j) * binomial(k,j) ) /k
nex j
next k
v = v*n
a(n) = v

That code DOES NOT result in expected value of a(11) = 815

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 <-- n
3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499, 2757, 5071, 9327 <-- a(n)

What is the correct interpretation of the equations from OEIS?

aidan, Oct 18, 2012