A370233 -id:A370233 - cp's OEIS frontend

A370234 a(n) = hypergeom([-n, 1 + n, 2 + n, -1 - n], [1/2, 1/2, 1], 1/16). Row sums of A370233.

1, 4, 24, 143, 881, 5534, 35213, 226238, 1464382, 9534594, 62377881, 409717004, 2700168229, 17845927498, 118238842174, 785086422265, 5222747607607, 34802614713382, 232261844100109, 1552139710253683, 10385180872126402, 69563083672190704, 466425975579323539

Offset: 0

Peter Luschny, Feb 13 2024

nonn

Cf. A370233.

Table[HypergeometricPFQ[{-1 - n, -n, 1 + n, 2 + n}, {1/2, 1/2, 1}, 1/16], {n, 0, 25}] (* Vaclav Kotesovec, Feb 18 2024 *)

a(n) = round(hypergeom([-n, 1 + n, 2 + n, -1 - n], [1/2, 1/2, 1], 1/16)); \\ Michel Marcus, Feb 13 2024

a(n) ~ phi^(4*n+4) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 18 2024

A370232 Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 36, 25, 1, 1, 100, 225, 49, 1, 1, 225, 1225, 784, 81, 1, 1, 441, 4900, 7056, 2025, 121, 1, 1, 784, 15876, 44100, 27225, 4356, 169, 1, 1, 1296, 44100, 213444, 245025, 81796, 8281, 225, 1, 1, 2025, 108900, 853776, 1656369, 1002001, 207025, 14400, 289, 1

Offset: 0

Peter Luschny, Feb 12 2024

			Triangle starts:
[0] 1;
[1] 1,     1;
[2] 1,     9,       1;
[3] 1,    36,      25,       1;
[4] 1,   100,     225,      49,       1;
[5] 1,   225,    1225,     784,      81,      1;
[6] 1,   441,    4900,    7056,    2025,    121,     1;
[7] 1,   784,   15876,   44100,   27225,   4356,   169,   1;

Shifted bisection of A182878.

Cf. A370233 (c=2), A188648 (row sums), A188662 (central terms).

Table[Binomial[n + k, 2*k]^2, {n, 0, 7}, {k, 0, n}] // Flatten

T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 1.

T(n, k) = [z^k] hypergeom([-n, -n, 1 + n, 1 + n], [1/2, 1/2, 1], z/16).

cp's OEIS Frontend

A370234 a(n) = hypergeom([-n, 1 + n, 2 + n, -1 - n], [1/2, 1/2, 1], 1/16). Row sums of A370233.

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