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A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

%I A376863 #7 Oct 12 2024 15:37:48
%S A376863 1,3,1,13,7,1,73,50,12,1,501,400,125,18,1,4051,3609,1335,255,25,1,
%T A376863 37633,36463,15214,3485,460,33,1,394353,408694,186949,48769,7805,763,
%U A376863 42,1,4596553,5036792,2479602,714364,131299,15708,1190,52,1,58941091,67714809,35419350,11045558,2256933,312375,29190,1770,63,1,824073141,986271823,543025851,180766890,40194965,6221397,676893,50970,2535,75,1
%N A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).
%H A376863 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. 19-12. In Russian.
%F A376863 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 = 1.
%e A376863 Triangle starts:
%e A376863 [0]        1;
%e A376863 [1]        3,        1;
%e A376863 [2]       13,        7,        1;
%e A376863 [3]       73,       50,       12,       1;
%e A376863 [4]      501,      400,      125,      18,       1;
%e A376863 [5]     4051,     3609,     1335,     255,      25,       1;
%e A376863 [6]    37633,    36463,    15214,    3485,     460,      33,      1;
%e A376863 [7]   394353,   408694,   186949,   48769,    7805,     763,     42,    1;
%e A376863 [8]  4596553,  5036792,  2479602,  714364,  131299,   15708,   1190,   52,     1;
%p A376863 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:=1:seq(seq(T(m,n,k),k=0..n),n=0..10);
%Y A376863 A000262 (column 0), A052852 (row sums).
%Y A376863 Triangle for m=0: A130534.
%K A376863 nonn,tabl
%O A376863 0,2
%A A376863 _Igor Victorovich Statsenko_, Oct 07 2024