A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

1, 3, 22, 223, 2792, 42671, 761984, 15707707, 365122688, 9491746747, 271962399232, 8539383210711, 290937486190592, 10710312199270503, 422984587596455936, 17864076455770831219, 802450164859200372736, 38242916911507537149427, 1925477163696152909447168, 102213291475268656299164879

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

Formula

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.

Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.

For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).

Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).

a(n) ~ n^n.

Example

Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Crossrefs

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341477.

Sequence in context: A141152 A372250 A066573 * A303190 A173142 A073530

Adjacent sequences: A341473 A341474 A341475 * A341477 A341478 A341479

Keyword

nonn

Author

Vaclav Kotesovec, Feb 13 2021

Status

approved