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
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
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.