Is this the correct use of nested sums & binomials?

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

  1. aidan

    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[0] = 3;


    Maintained by The OEIS Foundation Inc.
    Content is available under The OEIS End-User License Agreement .




    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
    #1
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