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Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).
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%I #11 Oct 17 2024 18:50:43

%S 1,5,1,32,11,1,248,113,18,1,2248,1230,263,26,1,23272,14534,3765,505,

%T 35,1,270400,186992,55654,9115,865,45,1,3479744,2612000,865186,163779,

%U 19110,1372,56,1,49079936,39434448,14235388,3013164,408569,36288,2058,68,1

%N Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

%C These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

%H Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-10-1.pdf#page=19">Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers</a>, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

%F T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.

%e [0] 1;

%e [1] 5, 1;

%e [2] 32, 11, 1;

%e [3] 248, 113, 18, 1;

%e [4] 2248, 1230, 263, 26, 1;

%e [5] 23272, 14534, 3765, 505, 35, 1;

%e [6] 270400, 186992, 55654, 9115, 865, 45, 1;

%e [7] 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1;

%e [8] 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1;

%p T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

%Y A361649 (row sums).

%Y Triangle for m=0: A130534.

%Y Triangle for m=1: A376863.

%K nonn

%O 0,2

%A _Igor Victorovich Statsenko_, Oct 14 2024