A336345 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2 * (exp(x) - Sum_{j=0..k} x^j/j!)).

1, 1, 2, 1, 0, 6, 1, 0, 2, 22, 1, 0, 0, 2, 94, 1, 0, 0, 2, 14, 454, 1, 0, 0, 0, 2, 42, 2430, 1, 0, 0, 0, 2, 2, 222, 14214, 1, 0, 0, 0, 0, 2, 42, 1066, 89918, 1, 0, 0, 0, 0, 2, 2, 142, 6078, 610182, 1, 0, 0, 0, 0, 0, 2, 2, 366, 36490, 4412798, 1, 0, 0, 0, 0, 0, 2, 2, 142, 3082, 238046, 33827974

Offset: 0

Author

Seiichi Manyama, Nov 20 2020

Keywords

• nonn
• tabl

Examples

			Square array begins:
     1,   1,  1, 1, 1, 1, 1, ...
     2,   0,  0, 0, 0, 0, 0, ...
     6,   2,  0, 0, 0, 0, 0, ...
    22,   2,  2, 0, 0, 0, 0, ...
    94,  14,  2, 2, 0, 0, 0, ...
   454,  42,  2, 2, 2, 0, 0, ...
  2430, 222, 42, 2, 2, 2, 0, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=0..4 give A001861, A194689, A339014, A339017, A339027.

Main diagonal gives A000007.

Cf. A293024.

Programs

PARI

{T(n, k) = n!*polcoef(prod(j=k+1, n, exp((x^j+x*O(x^n))/j!))^2, n)}

Ruby

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << 2 * (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
  ary
end
def A336345(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A336345(20)

Formula

E.g.f. of column k: (Product_{j>k} exp(x^j/j!))^2.

T(0,k) = 1, T(1,k) = T(2,k) = ... = T(k,k) = 0 and T(n,k) = 2 * Sum_{j=k..n-1} binomial(n-1,j)*T(n-1-j,k) for n > k.