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A383206 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * Stirling2(j,k).

%I A383206 #11 Apr 19 2025 10:04:05
%S A383206 1,0,1,0,3,1,0,11,9,1,0,49,71,18,1,0,257,575,245,30,1,0,1539,4957,
%T A383206 3120,625,45,1,0,10299,45829,39697,11480,1330,63,1,0,75905,454015,
%U A383206 517790,201677,33250,2506,84,1,0,609441,4804191,6999785,3513762,770007,81774,4326,108,1
%N A383206 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * Stirling2(j,k).
%F A383206 E.g.f. of column k (with leading zeros): (exp(f(x)) - 1)^k / k! with f(x) = (exp(2*x) - 1)/2.
%e A383206 Triangle starts:
%e A383206   1;
%e A383206   0,     1;
%e A383206   0,     3,     1;
%e A383206   0,    11,     9,     1;
%e A383206   0,    49,    71,    18,     1;
%e A383206   0,   257,   575,   245,    30,    1;
%e A383206   0,  1539,  4957,  3120,   625,   45,  1;
%e A383206   0, 10299, 45829, 39697, 11480, 1330, 63, 1;
%e A383206   ...
%o A383206 (PARI) T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 2));
%Y A383206 Columns k=0..3 give A000007, A004211 (for n > 0), A383207, A383208.
%Y A383206 Row sums give A380228.
%Y A383206 Cf. A130191.
%K A383206 nonn,tabl
%O A383206 0,5
%A A383206 _Seiichi Manyama_, Apr 19 2025