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- #!/usr/bin/ruby
- # See also:
- # https://oeis.org/A097893
- # https://mathworld.wolfram.com/CentralTrinomialCoefficient.html
- func central_trinomial_triangle(num, callback) {
- var row = [1]
- { |n|
- callback(row)
- row = [1, {|i| row[i-1] + row[i] + row[i+1] }.map(0 .. n-2)..., 1]
- } << 1..num
- }
- central_trinomial_triangle(12, {|row|
- row.map { "%4s" % _ }.join(' ').say
- })
- say "\n=> Partial sums of the central trinomial coefficients:"
- say gather {
- central_trinomial_triangle(15, {|row| take(row.sum) })
- }
- say gather {
- ^15 -> map { |n|
- take(0 .. n>>1 -> sum_by {|k|
- binomial(n + 1, 2*k + 1) * binomial(2*k, k)
- })
- }
- }
- __END__
- 1
- 1 1
- 1 3 1
- 1 5 5 1
- 1 7 11 11 1
- 1 9 19 29 23 1
- 1 11 29 57 71 53 1
- 1 13 41 97 157 181 125 1
- 1 15 55 151 295 435 463 307 1
- 1 17 71 221 501 881 1193 1205 771 1
- 1 19 89 309 793 1603 2575 3279 3169 1977 1
- 1 21 109 417 1191 2705 4971 7457 9023 8425 5147 1
- => Partial sums of the central trinomial coefficients:
- [1, 2, 5, 12, 31, 82, 223, 616, 1723, 4862, 13815, 39468, 113257, 326198, 942425]
- [1, 2, 5, 12, 31, 82, 223, 616, 1723, 4862, 13815, 39468, 113257, 326198, 942425]
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