A085362 -id:A085362 - cp's OEIS frontend

A085455 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-3)^n.

1, -2, 4, -10, 26, -70, 192, -534, 1500, -4246, 12092, -34606, 99442, -286730, 829168, -2403834, 6984234, -20331558, 59287740, -173149662, 506376222, -1482730098, 4346486256, -12754363650, 37461564504, -110125172682, 323990062452, -953883382354, 2810310510110, -8284915984726

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 01 2003

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Absolute values are in A025565.

Cf. A085362, A085363, A085364.

CoefficientList[Series[Sqrt[(1-x)/(1+3x)], {x, 0, 30}], x]

a(n) = sum(k=0, n, (-1)^k*binomial(n-1, n-k)*binomial(2*k, k)); \\ Seiichi Manyama, Feb 03 2023

G.f.: A(x)=Sqrt((1-x)/(1+3x)).
G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (x-1)*(4*k+2) - x*(x-1)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (x-1)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
From Seiichi Manyama, Feb 03 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^k * binomial(n-1,n-k) * binomial(2*k,k).
n*a(n) = -2*n*a(n-1) + 3*(n-2)*a(n-2). (End)
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A359489 Expansion of 1/sqrt(1 - 4*x/(1-x)^3).

Original entry on oeis.org

1, 2, 12, 68, 396, 2358, 14262, 87252, 538440, 3345434, 20899816, 131154264, 826135794, 5220372274, 33077821314, 210087769632, 1337104370320, 8525602760550, 54449281992528, 348250972411252, 2230296171922008, 14300414859019290, 91791793780179790

Offset: 0

Seiichi Manyama, Mar 24 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1218

Cf. A085362, A110170, A162478, A359758, A360132.

CoefficientList[Series[1/Sqrt[1-(4x)/(1-x)^3],{x,0,30}],x] (* Harvey P. Dale, Aug 09 2023 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x)^3))

a(n) = sum(k=0, n, binomial(2*k,k) * binomial(n+2*k-1,n-k)) \\ Winston de Greef, Mar 24 2023

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+2*k-1,n-k).
n*a(n) = (8*n-6)*a(n-1) - (10*n-24)*a(n-2) + 4*(n-3)*a(n-3) - (n-4)*a(n-4) for n > 3.
a(n) ~ sqrt(2*(2 + (35 + 3*sqrt(129))^(1/3))) * (40 + 7*(262 + 6*sqrt(129))^(1/3) + (262 + 6*sqrt(129))^(2/3))^n / ((43*(86 + 6*sqrt(129)))^(1/6) * sqrt(Pi*n) * 3^n * (262 + 6*sqrt(129))^(n/3)). - Vaclav Kotesovec, Mar 25 2023
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+1-k,2) * a(k). - Seiichi Manyama, Mar 28 2023

A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 16, 130, 1070, 8902, 74724, 631902, 5376840, 45990070, 395106656, 3407196982, 29477061166, 255733684010, 2224098916300, 19384492018770, 169270624419390, 1480625235653670, 12970844831940000, 113785067475668550, 999400688480388570

Offset: 0

Seiichi Manyama, Feb 03 2023

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A063886, A085362, A360317, A360318, A360319, A360322.

Table[Sum[5^(n-k) Binomial[n-1,n-k]Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jun 22 2025 *)

a(n) = sum(k=0, n, 5^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

my(N=30, x='x+O('x^N)); Vec(sqrt((1-5*x)/(1-9*x)))

G.f.: sqrt( (1-5*x)/(1-9*x) ).
n*a(n) = 2*(7*n-6)*a(n-1) - 45*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/5)^i * a(j) * a(i-j) = (9/5)^n.
a(n) ~ 2 * 3^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A377197 Expansion of 1/(1 - 4*x/(1-x))^(3/2).

Original entry on oeis.org

1, 6, 36, 206, 1146, 6258, 33728, 180018, 953628, 5021698, 26315676, 137350746, 714455826, 3705635646, 19171860336, 98973407550, 509963556330, 2623133951730, 13472299015580, 69098721151530, 353966981339070, 1811212435206070, 9258333786967920, 47281424213258070

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A085362, A377199, A377200.
Cf. A002457, A377198.
Cf. A374497.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(3/2))); // Vincenzo Librandi, May 11 2025

Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n-1, n-k));

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * a(k).
a(n) = (6*n*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n-1,n-k).
a(n) ~ 16 * sqrt(n) * 5^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Oct 26 2024
a(n) = 6*hypergeom([5/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 72, 264, 984, 3714, 14148, 54284, 209482, 812196, 3161340, 12345658, 48348522, 189807336, 746740510, 2943359208, 11620961412, 45950375602, 181936110006, 721233025332, 2862271873966, 11370584735100, 45212101270728, 179926167512914

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A085362, A360290, A360292.

Cf. A360186, A360294, A383581.

a(n) = sum(k=0, n\3, binomial(n-1-2*k, k)*binomial(2*n-6*k, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^3)))

G.f.: 1 / sqrt(1-4*x/(1-x^3)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-3)*a(n-3) - 2*(2*n-10)*a(n-4) - (n-6)*a(n-6).
a(n) = A383581(n) - A383581(n-3). - Seiichi Manyama, May 01 2025

A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

Original entry on oeis.org

1, 2, 6, 20, 70, 254, 936, 3492, 13150, 49882, 190318, 729576, 2807816, 10841962, 41983588, 162973568, 633994982, 2471010742, 9646981054, 37718873700, 147676286078, 578883674722, 2271704404900, 8923807316892, 35087269756344, 138075819924306

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A085362, A360290, A360291.

Cf. A360295, A383582.

a(n) = sum(k=0, n\4, binomial(n-1-3*k, k)*binomial(2*n-8*k, n-4*k));

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

G.f.: 1 / sqrt(1-4*x/(1-x^4)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-4)*a(n-4) - 2*(2*n-13)*a(n-5) - (n-8)*a(n-8).
a(n) = A383582(n) - A383582(n-4). - Seiichi Manyama, May 01 2025

A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 14, 100, 726, 5340, 39692, 297544, 2245990, 17050796, 130061412, 996078456, 7654571772, 58995989400, 455857911768, 3530234227344, 27392392806534, 212918339726028, 1657570714812020, 12922254685161112, 100867892292766612

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360318, A360321, A360322.

a(n) = sum(k=0, n, 4^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

my(N=30, x='x+O('x^N)); Vec(sqrt((1-4*x)/(1-8*x)))

G.f.: sqrt( (1-4*x)/(1-8*x) ).
n*a(n) = 2*(6*n-5)*a(n-1) - 32*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/4)^i * a(j) * a(i-j) = 2^n.
a(n) ~ 2^(3*n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 8^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, -4, 10, -30, 102, -376, 1462, -5900, 24470, -103644, 446382, -1948854, 8605290, -38362200, 172423770, -780496110, 3554991270, -16281079900, 74927379550, -346328465930, 1607078948690, -7483861047480, 34963419415650, -163825013554400, 769694347677002

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360318, A360319, A360321.

Cf. A007317.

a(n) = sum(k=0, n, (-5)^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

my(N=30, x='x+O('x^N)); Vec(sqrt((1+5*x)/(1+x)))

G.f.: sqrt( (1+5*x)/(1+x) ).
n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.
a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A361815 Expansion of 1/sqrt(1 - 4x(1-x)^2).

Original entry on oeis.org

1, 2, 2, -2, -14, -32, -30, 64, 346, 752, 584, -2044, -9486, -19324, -11368, 66180, 271658, 514916, 192584, -2151612, -7949736, -13933280, -1779028, 69933368, 235295106, 378579404, -61171228, -2267724644, -7003832456, -10248117752, 5236354188, 73288104568

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Diagonal of rational function 1/(1 - (1 - x*y) * (x + y)).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361816, A361817.

Cf. A137635.

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^2))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(2*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 2*(2*n-2)*a(n-2) + (2*n-3)*a(n-3) ) for n > 2.

A361816 Expansion of 1/sqrt(1 - 4x(1-x)^3).

Original entry on oeis.org

1, 2, 0, -10, -22, 12, 174, 344, -354, -3304, -5780, 9180, 65258, 99132, -226620, -1313580, -1690990, 5441340, 26681700, 28070100, -128211552, -543818824, -440381780, 2978145240, 11080939914, 6162798092, -68377892976, -225107280388, -64286124152

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A361834.
Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361817.
Cf. A361812.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^3))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(3*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) - (2*n-4)*a(n-4) ) for n > 3.

cp's OEIS Frontend

A085455 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-3)^n.

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A359489 Expansion of 1/sqrt(1 - 4*x/(1-x)^3).

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A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A377197 Expansion of 1/(1 - 4*x/(1-x))^(3/2).

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A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,k) * binomial(2*n-6*k,n-3*k).

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A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).

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A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

A361815 Expansion of 1/sqrt(1 - 4x(1-x)^2).

A361816 Expansion of 1/sqrt(1 - 4x(1-x)^3).