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 1-9 of 9 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

A187365 Trisection of A000984 (central binomial coefficients): binomial(2(3n+2),3n+2)/3!, n>=0.

Original entry on oeis.org

1, 42, 2145, 117572, 6686100, 388934370, 22974421470, 1372238454600, 82653088824684, 5011211083256840, 305437356823765089, 18697712969443807572, 1148770108115543559100, 70797430141465286938140, 4374750896947475198160300, 270950190057528375091435920
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

See a comment under A187357 concerning trisection.
This appears also in the trisection of A001700: binomial(2*(3*n+1)+1,(3*n+1)+1)/3.

Links

Crossrefs

Cf. A066802 binomial(6n,3n), A187364 binomial(2*(3n+1),3n+1)/2, A002458, A100033.

Formula

a(n)=binomial(2*(3*n+2),3*n+2)/3!, n>=0.
a(n)=binomial(3*(2*n+1),3*n+2)/3, n>=0.
O.g.f.:(cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1-4*x^(1/3))/2))/(18*x^(2/3)),
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) = (1/6)*Sum_{k = 0..3*n+2} binomial(3*n+2,k)^2.
a(n) = (1/6)*hypergeom([-2 - 3*n, -2 - 3*n], [1], 1).
a(n) = 8*(2*n + 1)*(6*n + 1)*(6*n - 1)/(n*(3*n + 1)*(3*n + 2)) * a(n-1). (End)

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

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

A187444 A trisection of A001405 (central binomial coefficients): binomial(3n+2,floor((3n+2)/2))/2, n>=0.

Original entry on oeis.org

1, 5, 35, 231, 1716, 12155, 92378, 676039, 5200300, 38779380, 300540195, 2268783825, 17672631900, 134564468610, 1052049481860, 8061900920775, 63205303218876, 486734856412028, 3824345300380220, 29566145391215356, 232714176627630544, 1804857108504066435
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

Cf. A187442: binomial(3*n,floor(3*n/2)), A187443: binomial(3*n+1,floor((3*n+1)/2)).

Programs

  • PARI
    vector(30, n, n--; binomial(3*n+2,(3*n+2)\2)/2) \\ Michel Marcus, Jun 11 2015

Formula

a(n) = binomial(3*n+2,floor((3*n+2)/2))/2, n>=0.
O.g.f.: G1(x^2) + x*G2(x^2), with G1(x) and G2(x) the o.g.f.s of A187364 and A187366, respectively.

A277170 Numerator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

Original entry on oeis.org

1, -1, 1, -1, 1, -3, 1, -1, 25, -1, 1, -49, 1, -1, 9, -3, 1, -363, 3025, -1, 169, -169, 1, -3, 1, -49, 289, -289, 7225, -361, 361, -361, 1, -1, 1, -529, 529, -529, 330625, -148225, 3025, -675, 9, -3, 7569, -2523, 142129, -409757907, 808201, -961, 8649, -2883, 1, -147
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2016

Keywords

Comments

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

References

  • Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Links

Crossrefs

Cf. A005810, A066802, A052203, A187364, A277520.

Programs

  • Mathematica
    a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Numerator; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Oct 22 2016 *)

Formula

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = a(n) / A277520(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

A277520 Denominator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

Original entry on oeis.org

1, 3, 25, 147, 1089, 20449, 48841, 312987, 55190041, 14322675, 100100025, 32065374675, 4546130625, 29873533563, 1859904071089, 4089135109921, 9399479144449, 22568149425822049, 1293753708921104809, 2835106739783283, 3289668853728536041
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2016

Keywords

Comments

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

References

  • Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Links

Crossrefs

Cf. A005810, A052203, A066802, A187364, A277170 (numerators).

Programs

  • Mathematica
    a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Denominator;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 22 2016 *)

Formula

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = A277170(n) / a(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

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 1-9 of 9 results.

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

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