motzkin_triangle.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 23 February 2018
# https://github.com/trizen

# Simple generation of the Motzkin triangle.

# See also:
#   https://oeis.org/A026300   (Motzkin triangle)
#   https://oeis.org/A005773   (sum of rows)
#   https://oeis.org/A129703   (sum of rows + 1)

# For n>0, A129703(n)-1 is the sum of the n-th row of Motzkin's triangle.

func motzkin_triangle(num, callback) {
    var row = []
    { |n|
        row = [0, 1, {|i| row[i+2] + row[i] + row[i+1] }.map(0 .. n-3)..., 0]
        callback(row.grep{ .> 0 })
    } << 2..(num+1)
}

motzkin_triangle(12, {|row|
    row.map { "%4s" % _ }.join(' ').say
})

say "\n=> Sum of the rows of Motzkin's triangle:"

say gather {
    motzkin_triangle(15, {|row| take(row.sum) })
}

func motzkin_row_sum(n) {
    sum(0 .. n, {|k|
        sum(0 .. floor(k/2), {|t|
             binomial(n, 2*t + n - k) * (binomial(2*t + n - k, t) - binomial(2*t + n - k, t-1))
        })
    })
}

say 15.of(motzkin_row_sum)

__END__
   1
   1    1
   1    2    2
   1    3    5    4
   1    4    9   12    9
   1    5   14   25   30   21
   1    6   20   44   69   76   51
   1    7   27   70  133  189  196  127
   1    8   35  104  230  392  518  512  323
   1    9   44  147  369  726 1140 1422 1353  835
   1   10   54  200  560 1242 2235 3288 3915 3610 2188
   1   11   65  264  814 2002 4037 6765 9438 10813 9713 5798

=> Sum of the rows of Motzkin's triangle:
[1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917]
[1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917]