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 1-5 of 5 results.

A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

Original entry on oeis.org

1, 7, 85, 1159, 16641, 246047, 3707509, 56610575, 872893441, 13560999991, 211939849045, 3328419072535, 52481589415425, 830317511708367, 13174519143904245, 209559710593266719, 3340604559333629953, 53354776911196959335, 853607938952248383829, 13677336690921351929767
Offset: 0

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Author

Clark Kimberling

Keywords

Comments

If the Delannoy triangle is defined by the Maple code in A008288, this is A008288(n, 3*n-2), n >= 1. - N. J. A. Sloane, Oct 29 2006

Links

Crossrefs

Cf. A008288, A001850, A026000, A331329.

Programs

  • Maple
    F := (2-t)/(3*t^2-8*t+2);  G := t*(t-1)^3/(t-2);  Ginv := RootOf(numer(G-x),t); ogf := series(eval(F, t=Ginv), x=0, 25); # Mark van Hoeij, Oct 30 2011
  • Mathematica
    a[n_] := Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, -4 n, -1];
Array[a, 20, 0] (* Peter Luschny, Jan 31 2020 *)

Formula

G.f.: F(G^(-1)(x)) where F = (2-t)/(3*t^2-8*t+2) and G = t*(t-1)^3/(t-2). - Mark van Hoeij, Oct 30 2011
From Peter Bala, Jan 29 2020: (Start)
a(n) = Sum_{k = 0..n} C(n,k)*C(3*n+k,n).
a(n) = Sum_{k = 0..n} C(n,k)*C(4*n-k,n).
a(n) = Sum_{k = 0..n} C(3*n,n-k)*C(3*n+k,k).
a(n) = Sum_{k = 0..n} 2^k*C(n,k)*C(3*n,k).
a(n) = Sum_{k = 0..n} C(4*n-k,k)*C(4*n-2*k,n-k).
3*n*(3*n - 1)*(3*n - 2)*(70*n^2 - 189*n + 127)*a(n) = 2*(15610*n^5 - 65562*n^4 + 102255*n^3 - 72864*n^2 + 23369*n - 2640)*a(n-1) - 3*(n - 1)* (3*n - 4)*(3*n - 5)*(70*n^2 - 49*n + 8)*a(n-2) with a(0) = 1, a(1) = 7.
(End)
a(n) = binomial(4*n, n)*hypergeom([-3*n, -n], [-4*n], -1). - Peter Luschny, Jan 31 2020
a(n) ~ sqrt(1 + 13/(4*sqrt(10))) * (223 + 70*sqrt(10))^n / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Feb 13 2021
D-finite with recurrence +435*n*(3*n-1)*(3*n-2)*a(n) +(-53978*n^3+43545*n^2+39923*n-35580)*a(n-1) +3*(-57648*n^3+321915*n^2-580787*n+339980)*a(n-2) +9*(1634*n^3-11365*n^2+27137*n-22546)*a(n-3) -27*(3*n-10)*(3*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 01 2022

Extensions

Corrected and extended by N. J. A. Sloane, Oct 29 2006

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

Views

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).

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 1-5 of 5 results.

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

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