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A367022
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Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).
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5
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1, 4, 1, 16, 12, 2, 64, 96, 48, 5, 256, 640, 640, 200, 14, 1024, 3840, 6400, 4000, 840, 42, 4096, 21504, 53760, 56000, 23520, 3528, 132, 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429, 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n, k) = 4^(n - k)*binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k + 1)^2.
T(n, k) = A001263(n+1, k+1)*4^(n - k)*binomial(2*k, k)/(k + 1). (End)
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EXAMPLE
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Triangle T(n, k) starts:
[0] 1;
[1] 4, 1;
[2] 16, 12, 2;
[3] 64, 96, 48, 5;
[4] 256, 640, 640, 200, 14;
[5] 1024, 3840, 6400, 4000, 840, 42;
[6] 4096, 21504, 53760, 56000, 23520, 3528, 132;
[7] 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429;
[8] 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430;
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MAPLE
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p := n -> 4^n*hypergeom([1/2, -n - 1, -n], [2, 2], x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..8);
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MATHEMATICA
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T[n_, k_]:=4^(n-k)*Binomial[n, k]*Binomial[n+1, k]*Binomial[2*k, k]/(k+1)^2; Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Nov 20 2023 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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