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.

A322244 G.f.: 1/sqrt(1 - 6*x - 55*x^2).

Original entry on oeis.org

1, 3, 41, 315, 3345, 31923, 328889, 3337323, 34600225, 359225955, 3760299081, 39497556123, 416692693041, 4409256847635, 46791791441625, 497734241873355, 5305782027097665, 56663444325365955, 606142658305541225, 6493612892317230075, 69658589316520324945, 748141936546712050035, 8043908203413946807545, 86573015247061060850475, 932597459464760512144225
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 41*x^2 + 315*x^3 + 3345*x^4 + 31923*x^5 + 328889*x^6 + 3337323*x^7 + 34600225*x^8 + 359225955*x^9 + 3760299081*x^10 + ...
such that A(x)^2 = 1/(1 - 6*x - 55*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 25*x^2 + 171*x^3 + 1457*x^4 + 12243*x^5 + 109769*x^6 + 997755*x^7 + 9314657*x^8 + 88177059*x^9 + 847159161*x^10 + ...

Links

Crossrefs

Cf. A322245 (a(n)^2).

Programs

  • Mathematica
    a[n_] := Sum[(-5)^(n-k) * 4^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
CoefficientList[Series[1/Sqrt[1-6x-55x^2],{x,0,40}],x] (* Harvey P. Dale, Aug 13 2024 *)
  • PARI
    /* Using generating function: */
{a(n) = polcoeff( 1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, (-5)^(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, 11^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
{a(n) = polcoeff( (1 + 3*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-5)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 3*x + 16*x^2)^n.
a(n) ~ 11^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 13 2018
D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +55*(-n+1)*a(n-2)=0. - R. J. Mathar, Jan 16 2020
a(n) = (1/4)^n * Sum_{k=0..n} (-5)^k * 11^(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.