A098482 -id:A098482 - cp's OEIS frontend

A376792 Expansion of 1/sqrt((1 - x^4)^2 - 4*x).

1, 2, 6, 20, 71, 258, 954, 3572, 13501, 51404, 196858, 757472, 2926097, 11341032, 44080770, 171755976, 670664951, 2623732322, 10281616176, 40350944112, 158573538071, 623930435834, 2457658576132, 9690467310480, 38244489565051, 151064227161784, 597165099484632

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

From Seiichi Manyama, Apr 30 2025: (Start)
Number of lattice paths from (0,0) to (n,n) using steps (1,0),(0,1),(4,4).
Diagonal of the rational function 1 / (1 - x - y - x^4*y^4). (End)

Cf. A001850, A349713, A376791.

Cf. A098482.

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x^4)^2-4*x))

a(n) = sum(k=0, n\4, binomial(2*n-7*k, k)*binomial(2*n-8*k, n-4*k));

a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-7*k,k) * binomial(2*n-8*k,n-4*k).

A098483 Expansion of 1/sqrt((1-x)^2-8x^4).

Original entry on oeis.org

1, 1, 1, 1, 5, 13, 25, 41, 85, 205, 473, 985, 2021, 4365, 9785, 21673, 46965, 101581, 222745, 492665, 1087237, 2388749, 5251065, 11587529, 25633045, 56697933, 125345113, 277283353, 614212133, 1361824525, 3020426681, 6700678377

Offset: 0

Paul Barry, Sep 10 2004

1/sqrt((1-x)^2-4rx^4) expands to sum{k=0..floor(n/2), binomial(n-2k,k)binomial(n-3k,k)r^k}

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A098480, A098482, A098484.

CoefficientList[Series[1/Sqrt[(1-x)^2-8*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

a(n) = sum(k=0, n\2, binomial(n-2*k, k)*binomial(n-3*k, k)*2^k) \\ Michel Marcus, Jul 24 2013

a(n)=sum{k=0..floor(n/2), binomial(n-2k, k)binomial(n-3k, k)2^k}.
D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 8*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ (1+sqrt(1+8*sqrt(2)))^n / (sqrt(33+10*sqrt(2)-sqrt(265+596*sqrt(2))) * sqrt(Pi*n) * 2^(n-3/2)). - Vaclav Kotesovec, Jun 23 2014

A098484 Expansion of 1/sqrt((1-x)^2-12x^4).

Original entry on oeis.org

1, 1, 1, 1, 7, 19, 37, 61, 145, 397, 979, 2107, 4591, 10915, 26857, 63649, 146347, 339751, 808885, 1936717, 4588705, 10803133, 25559287, 60893551, 145231309, 345462145, 821110051, 1955736379, 4668132067, 11146642903, 26605635949

Offset: 0

Paul Barry, Sep 10 2004

1/sqrt((1-x)^2-4rx^4) expands to sum{k=0..floor(n/2), binomial(n-2k,k)binomial(n-3k,k)r^k}.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A098481, A098482, A098483.

CoefficientList[Series[1/Sqrt[(1-x)^2-12*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

a(n)=sum{k=0..floor(n/2), binomial(n-2k, k)binomial(n-3k, k)3^k}.
D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 12*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ sqrt(3) * (1+sqrt(1+8*sqrt(3)))^n / (sqrt(49+10*sqrt(3)-sqrt(397+884*sqrt(3))) * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Jun 23 2014

A113180 Expansion of 1/sqrt((1-2x)^2-8x^4).

Original entry on oeis.org

1, 2, 4, 8, 20, 56, 160, 448, 1240, 3440, 9632, 27200, 77216, 219840, 627200, 1793024, 5136480, 14743232, 42390400, 122064640, 351951232, 1015990528, 2936079360, 8493340672, 24591589120, 71262291456, 206666232832, 599778166784

Offset: 0

Paul Barry, Oct 16 2005

In general, 1/sqrt((1-a*x)^2-4*b*x^4) expands to Sum_{k=0..floor(n/2)} C(n-2k,k)*C(n-3k,k)*b^k*a^(n-4k).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

Cf. A098482, A113179.

CoefficientList[Series[1/Sqrt[(1-2*x)^2-8*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

x='x+O('x^50); Vec(1/sqrt((1-2*x)^2 - 8*x^4)) \\ G. C. Greubel, Mar 17 2017

a(n) = Sum_{k=0..floor(n/2)} C(n-2k,k)*C(n-3k,k)*2^(n-3k).
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2) + 8*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ (1+sqrt(1+2*sqrt(2)))^n / (sqrt(6+5*sqrt(2)-sqrt(70+56*sqrt(2))) * sqrt(Pi*n)). - Vaclav Kotesovec, Jun 23 2014

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A376792 Expansion of 1/sqrt((1 - x^4)^2 - 4*x).

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A098483 Expansion of 1/sqrt((1-x)^2-8x^4).

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A098484 Expansion of 1/sqrt((1-x)^2-12x^4).

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A113180 Expansion of 1/sqrt((1-2*x)^2-8*x^4).

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A113180 Expansion of 1/sqrt((1-2x)^2-8x^4).