A359758 -id:A359758 - cp's OEIS frontend

A085362 a(0)=1; for n>0, a(n) = 25^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

1, 2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003

Number of bilateral Schroeder paths (i.e. lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at even (zero, positive or negative) levels. Example: a(2)=8 because we have UDUD, UUDD, UHD, UDDU and their reflections in the x-axis. First differences of A026375. - Emeric Deutsch, Jan 28 2004
From G. C. Greubel, May 22 2020: (Start)
This sequence is part of a class of sequences, for m >= 0, with the properties:
a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
(4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
This sequence is the case of m=1. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Bisection of A026392.
Cf. A000351 (5^n), A026375, A085363, A085364, A085455.
Cf. A110170, A162478, A359758, A360132.
Cf. A063886, A360317, A360318, A360319, A360321, A360322.
Essentially the same as A026387.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-5*x)) )); // G. C. Greubel, May 23 2020

a := n -> `if`(n=0,1,2*hypergeom([3/2, 1-n], [2], -4)):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Jan 30 2017

CoefficientList[Series[Sqrt[(1-x)/(1-5x)], {x, 0, 25}], x]

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-5*x))) \\ Joerg Arndt, May 10 2013

def A085362_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-5*x)) ).list()
A085362_list(30) # G. C. Greubel, May 23 2020

G.f.: sqrt((1-x)/(1-5*x)).
Sum_{i=0..n} (Sum_{j=0..i} a(j)*a(i-j)) = 5^n.
D-finite with recurrence: a(n) = (2*(3*n-2)*a(n-1)-5*(n-2)*a(n-2))/n; a(0)=1, a(1)=2. - Emeric Deutsch, Jan 28 2004
a(n) ~ 2*5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)* (2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(n-1,n-k). - Vladimir Kruchinin, May 30 2016
a(n) = 2*hypergeom([3/2, 1-n], [2], -4) for n>0. - Peter Luschny, Jan 30 2017
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A110170 First differences of the central Delannoy numbers (A001850).

Original entry on oeis.org

1, 2, 10, 50, 258, 1362, 7306, 39650, 217090, 1196834, 6634890, 36949266, 206549250, 1158337650, 6513914634, 36718533570, 207412854786, 1173779487810, 6653482333450, 37770112857074, 214694383882498, 1221832400430482, 6961037946938250, 39697830840765090, 226596964146630658

Offset: 0

Emeric Deutsch, Jul 14 2005

Number of Delannoy paths of length n that do not start with a (1, 1) step (a Delannoy path of length n is a path from (0, 0) to (n, n), consisting of steps E = (1, 0), N = (0, 1) and D = (1, 1)). Example: a(1) = 2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).

For n > 0: a(n) = A128966(2*n,n). - Reinhard Zumkeller, Jul 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A110169.

Cf. A085362, A162478, A359758, A360132.

a110170 0 = 1
a110170 n = a128966 (2 * n) n  -- Reinhard Zumkeller, Jul 20 2013

with(orthopoly): a:=proc(n) if n=0 then 1 else P(n,3)-P(n-1,3) fi end: seq(a(n),n=0..25);
a := n -> `if`(n=0, 1, 2*hypergeom([1 - n, -n], [1], 2)):
seq(simplify(a(n)), n=0..24); # Peter Luschny, May 22 2017

CoefficientList[Series[(1 - x)/Sqrt[1 - 6 * x + x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 16 2013

G.f.: (1-z)/sqrt(1-6*z+z^2).
a(n) = P_n(3) - P_{n-1}(3) (n >= 1), where P_j is j-th Legendre polynomial.
From Paul Barry, Oct 18 2009: (Start)
G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);
G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);
a(n) = Sum_{k = 0..n} (0^(n + k) + C(n + k - 1, 2k - 1)) * C(2k, k) = 0^n + Sum_{k = 0..n} C(n + k - 1, 2k - 1) * C(2k, k). (End)
D-finite with recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2^(-1/4)*(3+2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 18 2012
a(n) = A277919(2n,n). - John P. McSorley, Nov 23 2016
a(n) = 2*hypergeom([1 - n, -n], [1], 2) for n>0. - Peter Luschny, May 22 2017
D-finite with recurrence: n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Jan 15 2020
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n^2-k^2) * a(k). - Seiichi Manyama, Mar 28 2023
G.f.: Sum_{n >= 0} binomial(2*n, n)*x^n/(1 - x)^(2*n) = 1 + 2*x + 10*x^2 + 50*x^3 + .... - Peter Bala, Apr 17 2024

A360132 Expansion of 1/sqrt(1 - 4*x/(1-x)^6).

Original entry on oeis.org

1, 2, 18, 134, 1010, 7788, 60978, 482708, 3853338, 30964238, 250150176, 2029781310, 16530857930, 135051216620, 1106287906140, 9083459084364, 74734798117570, 615998603183550, 5085522355488150, 42045309424052250, 348067638153560040, 2884832348569699340

Offset: 0

Seiichi Manyama, Mar 24 2023

nonn

Cf. A085362, A110170, A162478, A359489, A359758.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x)^6))

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+5*k-1,n-k).
n*a(n) = (11*n-9)*a(n-1) - (25*n-60)*a(n-2) + 35*(n-3)*a(n-3) - 35*(n-4)*a(n-4) + 21*(n-5)*a(n-5) - 7*(n-6)*a(n-6) + (n-7)*a(n-7) for n > 6.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+4-k,5) * a(k).

A361791 Expansion of 1/sqrt(1 - 4*x/(1+x)^5).

Original entry on oeis.org

1, 2, -4, -10, 30, 72, -238, -580, 1970, 4910, -16734, -42750, 144600, 379000, -1264700, -3402480, 11160730, 30828070, -99168820, -281279030, 885931600, 2580541580, -7948885910, -23779051760, 71572652480, 219906488302, -646332447086, -2039738985238, 5850898295170

Offset: 0

Seiichi Manyama, Mar 24 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..2034

Cf. A006139, A137635, A360133, A361790, A361792.

Cf. A359758.

a[n_]:=(-1)^(n+1)Pochhammer[n,4]HypergeometricPFQ[{3/2,1-n,1+n/4,(5+n)/4, (6+n)/4, (7+n)/4}, {6/5,7/5,8/5,9/5,2}, 2^10/5^5]/12; Join[{1},Array[a,28]] (* Stefano Spezia, Jul 11 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^5))

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k)) \\ Winston de Greef, Mar 24 2023

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k).
n*a(n) = -( (2*n-4)*a(n-1) + (11*n-14)*a(n-2) + 20*(n-3)*a(n-3) + 15*(n-4)*a(n-4) + 6*(n-5)*a(n-5) + (n-6)*a(n-6) ) for n > 5.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+3-k,4) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,4)*hypergeom([3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4], [6/5, 7/5, 8/5, 9/5, 2], 2^10/5^5)/12 for n > 0. - Stefano Spezia, Jul 11 2024

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

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.

A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).

Original entry on oeis.org

1, 2, -2, -16, -10, 118, 304, -500, -3754, -2488, 30866, 83716, -135568, -1080972, -792876, 9090484, 25788118, -39325156, -335074520, -271779024, 2820643842, 8348113120, -11788972644, -107836934448, -96107852032, 900943403012, 2778574561276, -3596374190416

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361816.

Cf. A361813.

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

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

cp's OEIS Frontend

A085362 a(0)=1; for n>0, a(n) = 2*5^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

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A110170 First differences of the central Delannoy numbers (A001850).

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A360132 Expansion of 1/sqrt(1 - 4*x/(1-x)^6).

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A361791 Expansion of 1/sqrt(1 - 4*x/(1+x)^5).

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

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

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

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A361817 Expansion of 1/sqrt(1 - 4*x*(1-x)^4).

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A085362 a(0)=1; for n>0, a(n) = 25^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

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

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

A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).