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A380851 Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.

%I A380851 #25 Mar 02 2025 02:45:21
%S A380851 1,-2,4,-2,8,16,-4,24,96,64,-10,80,480,640,256,-28,280,2240,4480,3584,
%T A380851 1024,-84,1008,10080,26880,32256,18432,4096,-264,3696,44352,147840,
%U A380851 236544,202752,90112,16384,-858,13728,192192,768768,1537536,1757184,1171456,425984,65536
%N A380851 Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.
%H A380851 Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2025-02-1.pdf#page=11">Identification of Riordan generalizations of binomial coefficients</a>, Innovation science No 02-1, State Ufa, Aeterna Publishing House, 2025, pp. 11-18, see table 10. In Russian.
%F A380851 T(n,k) = Sum_{i=0..n-k} binomial(i+m, m)*binomial(n+1, n-k-i)*r^(2*n)*(-1)^(i), for m = 3/2 and r = 2.
%F A380851 From _Peter Luschny_, Feb 07 2025: (Start)
%F A380851 T(n,k) = r^(2*n)*JacobiP(n - k, 1 + k, m - 1 - n, -1).
%F A380851 T(n,k) = 4^n*binomial(n, k)*hypergeom([3/2, k - n], [k + 1], 1). (End)
%e A380851 Triangle starts:
%e A380851        k = 0      1       2        3        4        5       6
%e A380851   n=0:     1;
%e A380851   n=1:    -2,     4;
%e A380851   n=2:    -2,     8,     16;
%e A380851   n=3:    -4,    24,     96,      64;
%e A380851   n=4:   -10,    80,    480,     640,     256;
%e A380851   n=5:   -28,   280,   2240,    4480,    3584,    1024;
%e A380851   n=6:   -84,  1008,  10080,   26880,   32256,   18432,   4096;
%p A380851 T:=(m,r,n,k)->add(binomial(i+m,m)*binomial(n+1,n-k-i)*r^(2*n)*(-1)^(i),i=0..n-k): m:=3/2: r:=2: seq(print(seq(T(m,r,n,k), k=0..n)), n=0..10);
%t A380851 T[n_, k_] := 4^n Binomial[n, k] Hypergeometric2F1[3/2, k - n, k + 1, 1];
%t A380851 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* _Peter Luschny_, Feb 07 2025 *)
%o A380851 (SageMath)  # Using function riordan_array from A256893.
%o A380851 RA = riordan_array((1 - x)^(3/2 - 1), x/(1-x), 7)
%o A380851 for n in range(7): print(4^n * RA.row(n)[:n+1])  # _Peter Luschny_, Feb 28 2025
%Y A380851 Columns: A002420 (k=0); A240530 (k=1).
%Y A380851 Triangle for m=-3, r=1: A104713; for m=-2, r=1: A104712; for m=-1, r=1: A135278; for m=0, r=1: A007318; for m=1, r=1: A097805; for m=2, r=1: A159854.
%K A380851 sign,tabl
%O A380851 0,2
%A A380851 _Igor Victorovich Statsenko_, Feb 06 2025