A146534
a(n) = 4*C(2n,n) - 3*0^n.
Original entry on oeis.org
1, 8, 24, 80, 280, 1008, 3696, 13728, 51480, 194480, 739024, 2821728, 10816624, 41602400, 160466400, 620470080, 2404321560, 9334424880, 36300541200, 141381055200, 551386115280, 2153031497760, 8416395854880, 32933722910400, 128990414732400, 505642425751008, 1983674131792416
Offset: 0
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a[n_]:=4*Binomial[2n,n]-3*KroneckerDelta[n,0]; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)
A380851
Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.
Original entry on oeis.org
1, -2, 4, -2, 8, 16, -4, 24, 96, 64, -10, 80, 480, 640, 256, -28, 280, 2240, 4480, 3584, 1024, -84, 1008, 10080, 26880, 32256, 18432, 4096, -264, 3696, 44352, 147840, 236544, 202752, 90112, 16384, -858, 13728, 192192, 768768, 1537536, 1757184, 1171456, 425984, 65536
Offset: 0
Triangle starts:
k = 0 1 2 3 4 5 6
n=0: 1;
n=1: -2, 4;
n=2: -2, 8, 16;
n=3: -4, 24, 96, 64;
n=4: -10, 80, 480, 640, 256;
n=5: -28, 280, 2240, 4480, 3584, 1024;
n=6: -84, 1008, 10080, 26880, 32256, 18432, 4096;
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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);
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T[n_, k_] := 4^n Binomial[n, k] Hypergeometric2F1[3/2, k - n, k + 1, 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Feb 07 2025 *)
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# Using function riordan_array from A256893.
RA = riordan_array((1 - x)^(3/2 - 1), x/(1-x), 7)
for n in range(7): print(4^n * RA.row(n)[:n+1]) # Peter Luschny, Feb 28 2025
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