A369381 - OEIS

%I #13 Mar 03 2024 11:28:16

%S 1,0,1,0,3,7,0,6,60,90,0,10,310,1505,1701,0,15,1260,14490,46620,42525,

%T 0,21,4445,105875,716205,1727110,1323652,0,28,14280,653100,8162000,

%U 38623200

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

%C 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 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 T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

%e n\k  0      1       2        3        4       5

%e 0:    1

%e 1:    0      1

%e 2:    0      3       7

%e 3:    0      6      60       90

%e 4:    0     10     310     1505     1701

%e 5:    0     15    1260    14490    46620    42525

%p 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 Cf. A007820 (right diagonal).

%K nonn,tabl

%O 0,5

%A _Igor Victorovich Statsenko_, Jan 22 2024