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 1-4 of 4 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

A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

1, 3, 22, 223, 2792, 42671, 761984, 15707707, 365122688, 9491746747, 271962399232, 8539383210711, 290937486190592, 10710312199270503, 422984587596455936, 17864076455770831219, 802450164859200372736, 38242916911507537149427, 1925477163696152909447168, 102213291475268656299164879
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, A341477.

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.

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).

A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 41, 63, 1, 1, 9, 85, 377, 321, 1, 1, 11, 145, 1159, 3649, 1683, 1, 1, 13, 221, 2625, 16641, 36365, 8989, 1, 1, 15, 313, 4991, 50049, 246047, 369305, 48639, 1, 1, 17, 421, 8473, 118721, 982729, 3707509, 3800305, 265729, 1
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2021

Keywords

Examples

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    41,     85,    145,     221, ...
  1,   63,   377,   1159,   2625,    4991, ...
  1,  321,  3649,  16641,  50049,  118721, ...
  1, 1683, 36365, 246047, 982729, 2908411, ...

Links

Crossrefs

Columns k=0..5 give A000012, A001850, A026000, A026001, A331329, A341491.
Rows n=0..2 give A000012, A005408, A102083.
Main diagonal gives A181675(n+1).
Cf. A008288.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));
  • PARI
    T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));

Formula

T(n,k) = A008288(n,k*n).
T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n,j).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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