cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A331329 a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).

Original entry on oeis.org

1, 9, 145, 2625, 50049, 982729, 19665841, 398796225, 8166636545, 168502295625, 3497529199185, 72949645000065, 1527671538372225, 32100078290806665, 676451066002195825, 14290577765009652865, 302557549412667613185, 6417968867896642617225, 136371773642235542394385
Offset: 0

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Author

Peter Luschny, Jan 31 2020

Keywords

Comments

Special case of generalized Delannoy numbers (see cross-references):
T(n, k) = binomial(k*n, n)*hypergeom([(1-k)*n, -n], [-k*n], -1).

Links

Crossrefs

Cf. A001850 (k=2), A026000 (k=3), A026001 (k=4), this sequence (k=5), A341491 (k=6).
Cf. A008288, A181675, A341476, A341477.

Programs

  • Mathematica
    a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];
Array[a, 19, 0]

Formula

a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - Vaclav Kotesovec, Feb 13 2021

A341491 a(n) = binomial(6*n, n) * hypergeom([-5*n, -n], [-6*n], -1).

Original entry on oeis.org

1, 11, 221, 4991, 118721, 2908411, 72616013, 1837271615, 46943003137, 1208483403179, 31297149356221, 814471993937855, 21281058718583873, 557930580979801755, 14669716953106628781, 386675596518995000191, 10214494658006725840897, 270345191656309313382475
Offset: 0

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Author

Vaclav Kotesovec, Feb 13 2021

Keywords

Comments

In general, for k > 1, binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1) ~ sqrt((k + (k^2 - k + 1) / sqrt(k^2 - 2*k + 2)) / (4*(k-1)*Pi*n)) * ((A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1))^n.

Links

Crossrefs

Cf. A001850, A026000, A026001, A331329, A341476, A341477.

Programs

  • Mathematica
    Table[Binomial[6*n, n] * Hypergeometric2F1[-5*n, -n, -6*n, -1], {n,0,20}]

Formula

a(n) ~ sqrt((6 + 31/sqrt(26))/(20*Pi*n)) * (42671 + 8346*sqrt(26))^n / 5^(5*n).

A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

0, 2, 10, 70, 680, 8346, 125504, 2218350, 45335680, 1047314578, 27079557632, 772687787510, 24172386314240, 821114930966890, 30146801401143296, 1187943632192716894, 50068690149298438144, 2245175953053786221730, 106828553482726336102400, 5371204894269759411503910
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 13 2021

Keywords

Examples

			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.

Links

Crossrefs

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341476.

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-1).
Showing 1-3 of 3 results.

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