A299506 -id:A299506 - cp's OEIS frontend

A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

1, 1, 1, 1, 3, 25, 1, 5, 43, 425, 1, 7, 65, 661, 7025, 1, 9, 91, 965, 10515, 116625, 1, 11, 121, 1345, 15105, 171097, 1951625, 1, 13, 155, 1809, 20995, 243525, 2828101, 32903225, 1, 15, 193, 2365, 28401, 337877, 4001345, 47284251, 558265825

Offset: 0

Peter Luschny, Mar 16 2018

			[0] 1,  1,  25,  425,  7025, 116625,  1951625,  32903225, ... [A299845]
[1] 1,  3,  43,  661, 10515, 171097,  2828101,  47284251, ... [A299506]
[2] 1,  5,  65,  965, 15105, 243525,  4001345,  66622085, ...
[3] 1,  7,  91, 1345, 20995, 337877,  5544709,  92234527, ... [A243946]
[4] 1,  9, 121, 1809, 28401, 458649,  7544041, 125700129, ... [A084769]
[5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947]
[6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...

Cf. A299845, A299506, A243946, A084769, A243947.

Cf. A300946.

Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}];
Table[Print[Arow[n, 7]], {n, 0, 6}];
T[n_, k_] := If[k==0, 1, 4^k*Sum[(5/4)^j*Binomial[k, j]*Binomial[k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)] ,{j, 0, n}]]; Flatten[Table[T[n, k],{n, 0, 8}, {k, 0, n}]] (* Detlef Meya, May 28 2024 *)

T(n, k) = if k = 0 then 1, otherwise 4^k*Sum_{j=0..n} (5/4)^j * binomial(k, j) * binomial(k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)). - Detlef Meya, May 28 2024

A299845 a(n) = hypergeom([-n, n - 1], [1], -4).

Original entry on oeis.org

1, 1, 25, 425, 7025, 116625, 1951625, 32903225, 558265825, 9522632225, 163160773625, 2806202183625, 48420275891025, 837813745045425, 14531896733426025, 252593595973313625, 4398859688478578625, 76733590756134492225, 1340547988367851940825, 23451231922182584693225

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..798

Cf. A299506, A243946, A084769, A243947.

Cf. A300945, A300946.

f:= gfun:-rectoproc({4*n*(n-2)^2*a(n)+4*(n-1)^2*(n-3)*a(n-2)-4*(2*n-3)*(9*n^2-27*n+17)*a(n-1)=0,
a(0)=1,a(1)=1,a(2)=25},a(n),remember):
map(f, [$0..100]); # Robert Israel, Mar 21 2018

a[n_] := Hypergeometric2F1[-n, n - 1, 1, -4]; Table[a[n], {n, 0, 19}]
a[0]:=1; a[1]:=1; a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, May 22 2024 *)

4*n*(n-2)^2*a(n) + 4*(n-1)^2*(n-3)*a(n-2) - 4*(2*n-3)*(9*n^2-27*n+17)*a(n-1) = 0.  - Robert Israel, Mar 21 2018
a(n) ~ 2^(-3/2) * 5^(3/4) * phi^(6*n - 3) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 05 2018
a(n) = 4^n*Sum_{k=0..n} (5/4)^k*Gamma(n + 1)*Gamma(n - 1)/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1)) for n >= 2. - Detlef Meya, May 22 2024

cp's OEIS Frontend

A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

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