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A369381 Triangle of numbers read by rows T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

%I A369381 #13 Mar 03 2024 11:28:16
%S A369381 1,0,1,0,3,7,0,6,60,90,0,10,310,1505,1701,0,15,1260,14490,46620,42525,
%T A369381 0,21,4445,105875,716205,1727110,1323652,0,28,14280,653100,8162000,
%U A369381 38623200
%N A369381 Triangle of numbers read by rows T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).
%C A369381 The triangle T(n,k) is a functional dual of the triangle A269939 in identity: B(n) = Sum_{k=0..n}(-1)^(k)*A269939(n,k)/Binomial(n+k,k) = Sum_{k=0..n}(-1)^(k)*T(n,k)/Binomial(n+k,k). Where B(n) are the Bernoulli numbers.
%H A369381 I. V. Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-01-2.pdf#page=15">Functional twin of triangle A269939</a>, Innovation science No 1-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
%F A369381 T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).
%e A369381 n\k  0      1       2        3        4       5
%e A369381 0:    1
%e A369381 1:    0      1
%e A369381 2:    0      3       7
%e A369381 3:    0      6      60       90
%e A369381 4:    0     10     310     1505     1701
%e A369381 5:    0     15    1260    14490    46620    42525
%p A369381 T:=(n,k)->((n+1)!/((k+1)!*(n-k)!))*Stirling2(n+k,k):seq(seq T(n,k),k=0..n), n=0..10);
%Y A369381 Cf. A007820 (right diagonal).
%K A369381 nonn,tabl
%O A369381 0,5
%A A369381 _Igor Victorovich Statsenko_, Jan 22 2024