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.

A322248 G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).

Original entry on oeis.org

1, 5, 57, 605, 6961, 81525, 973545, 11765325, 143522145, 1763351525, 21789466777, 270509191485, 3371353189009, 42155188480085, 528587607974217, 6644129071092525, 83691484792766145, 1056178325362832325, 13351036742005533945, 169019946403985898525, 2142600388730167543281, 27193744661180635582005, 345520219114720175821737, 4394534009569783690837005, 55943630366450131877449761, 712778930909503993783945125
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2018

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 39*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...

Links

Crossrefs

Cf. A084605, A098409, A248168.
Cf. A322249.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)
  • PARI
    /* Using generating function: */
{a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* a(n) as a central coefficient */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 5*x + 16*x^2)^n.
a(n) ~ 13^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 10 2018
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) + 39*(n-1)*a(n-2) for n > 1. - Seiichi Manyama, Apr 22 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A307810 Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).

Original entry on oeis.org

1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2019

Keywords

Comments

See A246923.
Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).

Links

Crossrefs

Cf. A307695.
(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}

Formula

a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.
a(n) ~ 2^(8*n+2) / (3*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A307811 Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).

Original entry on oeis.org

1, 49, 2601, 148225, 8970025, 570111129, 37678303881, 2567836387809, 179267329355625, 12754414737348025, 921185098227422161, 67340346156989933769, 4971327735657992896201, 369994703739586257235225, 27725052308247030792515625, 2089567204521186409129541025
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2019

Keywords

Comments

See A246923.
Also the squares of coefficients in g.f. 1/sqrt((1-5*x)*(1-9*x)).

Links

Crossrefs

Cf. A104454.
(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), this sequence (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), A307810 (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[5^(n-k) * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 16, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))
  • PARI
    {a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}
  • PARI
    {a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}

Formula

a(n) = A104454(n)^2 = (Sum_{k=0..n} 5^(n-k)*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 9^(n-k)*(-1)^k*binomial(n,k)*binomial(2k,k))^2.
a(n) ~ 3^(4*n+2) / (4*Pi*n). - Vaclav Kotesovec, Sep 27 2019
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.