9 billion names of God the integer

var cache = [[1]]
 
func cumu (n) {
    for l (cache.len .. n) {
        var r = [0]
        for i (1..l) {
            r << (r[-1] + cache[l-i][min(i, l-i)])
        }
        cache << r
    }
    cache[n]
}
 
func row (n) {
    var r = cumu(n)
    n.of {|i| r[i+1] - r[i] }
}
 
say "rows:"
for i (1..15) {
    "%2s: %s\n".printf(i, row(i))
}
 
say "\nsums:"
 
for i in [23, 123, 1234, 12345] {
    "%2s : %4s\n".printf(i, cumu(i)[-1])
}

Output:

rows:
 1: [1]
 2: [1, 1]
 3: [1, 1, 1]
 4: [1, 2, 1, 1]
 5: [1, 2, 2, 1, 1]
 6: [1, 3, 3, 2, 1, 1]
 7: [1, 3, 4, 3, 2, 1, 1]
 8: [1, 4, 5, 5, 3, 2, 1, 1]
 9: [1, 4, 7, 6, 5, 3, 2, 1, 1]
10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
11: [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1]
12: [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1]
13: [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1]
14: [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1]
15: [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1]

sums:
23 : 1255
123 : 2552338241
1234 : 156978797223733228787865722354959930
^C