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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!)).

%I A336345 #33 Nov 21 2020 09:10:25
%S A336345 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,
%T A336345 0,0,2,2,222,14214,1,0,0,0,0,2,42,1066,89918,1,0,0,0,0,2,2,142,6078,
%U A336345 610182,1,0,0,0,0,0,2,2,366,36490,4412798,1,0,0,0,0,0,2,2,142,3082,238046,33827974
%N 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!)).
%H A336345 Seiichi Manyama, <a href="/A336345/b336345.txt">Antidiagonals n = 0..139, flattened</a>
%F A336345 E.g.f. of column k: (Product_{j>k} exp(x^j/j!))^2.
%F A336345 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.
%e A336345 Square array begins:
%e A336345      1,   1,  1, 1, 1, 1, 1, ...
%e A336345      2,   0,  0, 0, 0, 0, 0, ...
%e A336345      6,   2,  0, 0, 0, 0, 0, ...
%e A336345     22,   2,  2, 0, 0, 0, 0, ...
%e A336345     94,  14,  2, 2, 0, 0, 0, ...
%e A336345    454,  42,  2, 2, 2, 0, 0, ...
%e A336345   2430, 222, 42, 2, 2, 2, 0, ...
%o A336345 (PARI) {T(n, k) = n!*polcoef(prod(j=k+1, n, exp((x^j+x*O(x^n))/j!))^2, n)}
%o A336345 (Ruby)
%o A336345 def ncr(n, r)
%o A336345   return 1 if r == 0
%o A336345   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A336345 end
%o A336345 def A(k, n)
%o A336345   ary = [1]
%o A336345   (1..n).each{|i| ary << 2 * (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
%o A336345   ary
%o A336345 end
%o A336345 def A336345(n)
%o A336345   a = []
%o A336345   (0..n).each{|i| a << A(i, n - i)}
%o A336345   ary = []
%o A336345   (0..n).each{|i|
%o A336345     (0..i).each{|j|
%o A336345       ary << a[i - j][j]
%o A336345     }
%o A336345   }
%o A336345   ary
%o A336345 end
%o A336345 p A336345(20)
%Y A336345 Columns k=0..4 give A001861, A194689, A339014, A339017, A339027.
%Y A336345 Main diagonal gives A000007.
%Y A336345 Cf. A293024.
%K A336345 nonn,tabl
%O A336345 0,3
%A A336345 _Seiichi Manyama_, Nov 20 2020