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 1-7 of 7 results.

A002458 a(n) = binomial(4*n+1, 2*n).

Original entry on oeis.org

1, 10, 126, 1716, 24310, 352716, 5200300, 77558760, 1166803110, 17672631900, 269128937220, 4116715363800, 63205303218876, 973469712824056, 15033633249770520, 232714176627630544, 3609714217008132870, 56093138908331422716, 873065282167813104916
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			1 + 10*x + 126*x^2 + 1716*x^3 + 24310*x^4 + 352716*x^5 + 5200300*x^6 + ...

References

  • The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1982, (3.109), page 35.

Links

Crossrefs

Cf. A000984, A001448, A001700, A024492, A100033, A187364, A187365.
Row sums of A067001.

Programs

  • Maple
    A002458:=n->binomial(4*n+1,2*n): seq(A002458(n), n=0..30); # Wesley Ivan Hurt, Jan 17 2017
  • Mathematica
    Table[Binomial[4n+1,2n],{n,0,30}] (* Harvey P. Dale, Apr 04 2011 *)
4^Range[0, 22] Simplify[ CoefficientList[ Series[ Sqrt[2]/(((Sqrt[1 - 4 x] + 1)^(1/2))*Sqrt[1 - 4 x]), {x, 0, 22}], x]] (* Robert G. Wilson v, Aug 08 2011 *)
  • PARI
    a(n) = binomial( 4*n + 1, 2*n)

Formula

a(n) = Sum_{k=0..n} 4^k * binomial( n + k, n) * binomial( 2*n - 2*k, n - k). - Michael Somos, Feb 25 2012
a(n) = A001700(2*n) = (n+1)*A000108(2*n+1).
G.f.: (4 - (1+4*y)*c(y) - (1-4*y)*c(-y))/(2*(1 - (4*y)^2)) with y^2 = x, c(y) = g.f. for A000108 (Catalan). - Wolfdieter Lang, Dec 13 2001
a(n) ~ 2^(1/2)*Pi^(-1/2)*n^(-1/2)*2^(4*n)*{1 - 5/16*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
a(n) = A024492(n)*(n+1). - R. J. Mathar, Aug 10 2015
G.f.: 2F1(3/4,5/4; 3/2; 16*x). - R. J. Mathar, Aug 10 2015
D-finite with recurrence n*(2*n + 1)*a(n) - 2*(4*n - 1)*(4*n + 1)*a(n-1) = 0. - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
a(n) = 4^n*binomial(2*n + 1/2, n).
O.g.f.: sqrt(c(4*x)/(1 - 16*x)) = sqrt(2/(1 - 16*x)/(1 + sqrt(1 - 16*x))), where
c(y) = g.f. for A000108 (Catalan). In general, c(x)^k/sqrt(1 - 4*x) is the o.g.f. for the sequence binomial(2*n + k, n). (End) [Edited by Petros Hadjicostas, May 25 2020]
From Ilya Gutkovskiy, Jan 17 2017: (Start)
E.g.f.: 2F2(3/4,5/4; 1,3/2; 16*x).
Sum_{n>=0} 1/a(n) = 3F2(1,1,3/2; 3/4,5/4; 1/16) = 1.108563435104316693... (End)
From Peter Bala, Mar 16 2018: (Start)
The right-hand side of the binomial coefficient identity Sum_{k = 0..n} 4^(n-k) * C(2*n+1, 2*k) * C(2*k, k) = a(n).
a(n) = 4^n*hypergeom([-n, -n-1/2], [1], 1). (End)
From Peter Bala, Mar 20 2023: (Start)
a(n) = Sum_{k = 0..n} binomial(2*n+1,k)^2.
a(n) = (1/2)*hypergeom([-1 - 2*n, -1 - 2*n], [1], 1). (End)

A066802 a(n) = binomial(6*n,3*n).

Original entry on oeis.org

1, 20, 924, 48620, 2704156, 155117520, 9075135300, 538257874440, 32247603683100, 1946939425648112, 118264581564861424, 7219428434016265740, 442512540276836779204, 27217014869199032015600, 1678910486211891090247320, 103827421287553411369671120, 6435067013866298908421603100
Offset: 0

Views

Author

Benoit Cloitre, Jan 18 2002

Keywords

Comments

For the trisection of a sequence (here A000984) given by its real o.g.f. see a comment and a reference under A187357.

Links

Crossrefs

Cf. A001450, A066798, A187364 (binomial(2(3n+1),3n+1)/2), A187365 (binomial(2(3n+2),3n+2)/3!).

Programs

  • Magma
    [Binomial(6*n, 3*n): n in [0..15]]; // G. C. Greubel, Feb 17 2020
  • Maple
    a := n -> hypergeom([-3*n, -3*n], [1], 1):
seq(simplify(a(n)), n=0..13); # Peter Luschny, Mar 19 2018
  • Mathematica
    Table[Binomial[6n, 3n], {n,0,13}] (* Jean-François Alcover, Jun 03 2019 *)
  • PARI
    a(n) = { binomial(6*n, 3*n) } \\ Harry J. Smith, Mar 28 2010
  • Sage
    [binomial(6*n, 3*n) for n in (0..15)] # G. C. Greubel, Feb 17 2020

Formula

a(n) = A000984(3*n).
a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} binomial(n, i)*binomial(n, j) *binomial(n, k)*binomial(3n, i+j+k). - Benoit Cloitre, Mar 08 2005
O.g.f. (with a(0):=1): (cb(x^(1/3)) + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))+1+2*x^(1/3)))/3, with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x) = 1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z) the o.g.f. of the Legendre polynomials). - Wolfdieter Lang, Mar 24 2011
D-finite with recurrence n*(3n-1)*(3n-2)*a(n) = 8*(6n-5)*(6n-1)*(2n-1)*a(n-1). - R. J. Mathar, Sep 17 2012
a(n) = GegenbauerC(3*n, -3*n, -1). - Peter Luschny, May 07 2016
a(n) = hypergeom([-3*n, -3*n], [1], 1). - Peter Luschny, Mar 19 2018
a(n) ~ 2^(6*n)/sqrt(3*Pi*n). - Vaclav Kotesovec, Jun 07 2019
From Peter Bala, Feb 16 2020: (Start)
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k.
a(n) = [(x*y)^(3*n)] (1 + x + y)^(6*n). Cf. A001448. (End)
Conjecture: a(n) = [x^n] G(x)^(2*n), where G(x) = (1 + x)*(1 - 6*x + x^2)/(2*x) + (x^2 - 1)*sqrt(1 - 14*x + x^2)/(2*x) = 1 + 10*x + 81*x^2 + 720*x^3 + .... The algebraic function G(x) satisfies the quadratic equation x*G(x)^2 - (1 - 5*x - 5*x^2 + x^3)*G(x) + (1 + x)^4 = 0. Cf. A001450. - Peter Bala, Oct 27 2022
a(n) = Sum_{k = 0..3*n} binomial(3*n+k-1, k). - Peter Bala, Jun 04 2024
O.g.f: 3F2(1/6,1/2,5/6; 1/3,2/3 ; 64*x). - R. J. Mathar, Jan 11 2025

Extensions

Extended to a(0)=1 by M. F. Hasler, Oct 06 2014

A187364 Trisection of A000984 (central binomial coefficients): binomial(2(3n+1),3n+1)/2, n>=0.

Original entry on oeis.org

1, 35, 1716, 92378, 5200300, 300540195, 17672631900, 1052049481860, 63205303218876, 3824345300380220, 232714176627630544, 14226520737620288370, 873065282167813104916, 53753604366668088230810, 3318776542511877736535400, 205397724721029574666088520
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

See a comment under A187363 concerning trisection.
This appears also in the trisection of A001700 (central binomials in the odd numbered Pascal rows): binomial(2*(3*n)+1,3*n+1).

Links

Crossrefs

Cf. A066802 (binomial(6n,3n)), A187365 (binomial(2(3n+2),3n+2)/3!).
Cf. A002458, A100033.

Programs

  • Mathematica
    Table[c=3n+1;Binomial[2c,c]/2,{n,0,20}] (* Harvey P. Dale, May 10 2012 *)

Formula

a(n) = binomial(2*(3*n+1),3*n+1)/2, n>=0.
a(n) = binomial(2*(3*n)+1,3*n+1), n>=0.
O.g.f.: (cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1+8*x^(1/3))/2))/(6*x^(1/3)), with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x)=1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z) the o.g.f. of the Legendre polynomials).
From Peter Bala, Mar 19 2023: (Start)
a(n) = (1/2)*Sum_{k = 0..3*n+1} binomial(3*n+1,k)^2.
a(n) = (1/2)*hypergeom([-1 - 3*n, -1 - 3*n], [1], 1).
a(n) = 8*(2*n - 1)*(6*n + 1)*(6*n - 1)/(n*(3*n + 1)*(3*n - 1)) * a(n-1). (End)
Right-hand side of the binomial sum identity (1/18) * Sum_{k = 0..6*n+3} (-1)^(n+k) * (k/(2*n + 1))^2 * binomial(6*n+3, k)^2 = a(n). - Peter Bala, Nov 05 2024

A100033 Bisection of A001700.

Original entry on oeis.org

3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370
Offset: 0

Views

Author

N. J. A. Sloane, Nov 20 2004

Keywords

Crossrefs

Cf. A001700, A002458, A187364, A187365.

Programs

  • Maple
    a:=n->binomial(4*n+3,2*n+2): seq(a(n),n=0..19);

Formula

a(n) = binomial(4*n+3, 2*n+2). - Emeric Deutsch, Dec 09 2004
From Peter Bala, Mar 19 2023: (Start)
a(n) = (1/2)*Sum_{k = 0..2*n+2} binomial(2*n+2,k)^2.
a(n) = (1/2)*hypergeom([-2 - 2*n, -2 - 2*n], [1], 1).
a(n) = 2*(4*n + 1)*(4*n + 3)/((n + 1)*(2*n + 1)) * a(n-1). (End)
From Peter Bala, Mar 28 2023: (Start)
a(n) = (1/(2*n + 2))*Sum_{k = 0..2*n+2} k*binomial(2*n+2,k)^2.
a(n) = 2*(n + 1)*hypergeom([-1 - 2*n, -1 - 2*n], [2], 1). (End)

Extensions

More terms from Emeric Deutsch, Dec 09 2004

A187442 A trisection of A001405 (central binomial coefficients): binomial(3*n,floor(3*n/2)), n>=0.

Original entry on oeis.org

1, 3, 20, 126, 924, 6435, 48620, 352716, 2704156, 20058300, 155117520, 1166803110, 9075135300, 68923264410, 538257874440, 4116715363800, 32247603683100, 247959266474052, 1946939425648112, 15033633249770520, 118264581564861424
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For the trisection of sequences see a comment and a reference under A187357.

Crossrefs

A187443 binomial(3*n+1,floor((3*n+1)/2)),
A187444 binomial(3*n+2,floor((3*n+2)/2))/2.

Programs

  • Mathematica
    Table[Binomial[3n,Floor[(3n)/2]],{n,0,20}] (* Harvey P. Dale, Dec 23 2012 *)

Formula

a(n) = binomial(3*n,floor(3*n/2)), n>=0.
O.g.f.: G0(x^2) + 3*x*G2(x^2) with G0(x) and G2(x) the o.g.f.s of A187363 and A187365, respectively.

A187443 A trisection of A001405 (central binomial coefficients): binomial(3n+1,floor((3n+1)/2)), n>=0.

Original entry on oeis.org

1, 6, 35, 252, 1716, 12870, 92378, 705432, 5200300, 40116600, 300540195, 2333606220, 17672631900, 137846528820, 1052049481860, 8233430727600, 63205303218876, 495918532948104, 3824345300380220, 30067266499541040, 232714176627630544, 1832624140942590534, 14226520737620288370, 112186277816662845432, 873065282167813104916
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For trisection of sequences see a comment and a reference under A187357.

Crossrefs

Cf. A187442: binomial(3n,floor(3n/2)), A187444: binomial(3n+2,floor((3n+2)/2))/2.
Cf. A187357, A187364, A187365.
Cf. A001405.

Programs

  • Mathematica
    Table[Binomial[3n+1,Floor[(3n+1)/2]],{n,0,30}] (* Harvey P. Dale, Jan 13 2021 *)

Formula

a(n) = binomial(3*n+1,floor((3*n+1)/2)), n>=0.
O.g.f.: 3!*x*G2(x^2) + G1(x^2), with G2(x) and G1(x) the o.g.f.s of A187365 and A187364, respectively.

Extensions

Corrected and extended by Harvey P. Dale, Jan 13 2021

A187366 One half of a trisection of A001700: binomial(6n+5,3(n+1))/2, n>=0.

Original entry on oeis.org

5, 231, 12155, 676039, 38779380, 2268783825, 134564468610, 8061900920775, 486734856412028, 29566145391215356, 1804857108504066435, 110628135069209194801, 6804253717299758003900, 419727621552972772561830, 25956855321888352842417780
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For trisection of a sequence see a comment and a reference under A187357.

Crossrefs

Cf. A187364 binomial(2(3n)+1,3n+1),
A187365 binomial(2(3n+1)+1,(3n+1)+1)/3.

Formula

a(n)= binomial(2*(3*n+2)+1,(3*n+2)+1)/2 = binomial(6*n+5,3*(n+1))/2 , n>=0.
O.g.f.: (cb(x^(1/3)) - 3 + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3)) + 1 + 2*x^(1/3)))/(12*x),
with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x)=1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z)the o.g.f. of the Legendre polynomials).
Showing 1-7 of 7 results.

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

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