A360318 -id:A360318 - 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)

A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 10, 52, 278, 1516, 8388, 46920, 264678, 1503052, 8581676, 49215256, 283297660, 1635904376, 9472214344, 54975423504, 319729353606, 1862896455180, 10871759717916, 63539265366264, 371837338366740, 2178604586281128, 12778264475444280, 75022726995053808

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360318, A360319, A360321, A360322.

Cf. A005573, A085363.

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

my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)))

G.f.: sqrt( (1-2*x)/(1-6*x) ).
n*a(n) = 2*(4*n-3)*a(n-1) - 12*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/2)^i * a(j) * a(i-j) = 3^n.
a(n) = 2 * A005573(n-1) for n > 0.
a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/2)^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} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

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)

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)

A383948 Expansion of 1/sqrt((1-3x)^3 (1-7*x)).

Original entry on oeis.org

1, 8, 51, 308, 1855, 11340, 70665, 448320, 2887155, 18815240, 123759097, 819969276, 5464090177, 36580917716, 245837438055, 1657396783440, 11204207037315, 75918595916520, 515462211835305, 3506072423912940, 23885410548196701, 162951783575205108, 1113110415733083531

Offset: 0

Seiichi Manyama, Aug 19 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A098409, A360318.
Cf. A383949, A383950.
Cf. A132894.

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 3*x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

CoefficientList[Series[ 1/Sqrt[(1-3*x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-3*x)^3*(1-7*x)))

n*a(n) = (10*n-2)*a(n-1) - 21*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

A387212 Expansion of sqrt((1-3x) / (1-7x)^3).

Original entry on oeis.org

1, 9, 75, 599, 4659, 35595, 268485, 2005785, 14873715, 109643195, 804354417, 5877232773, 42798735805, 310767250773, 2250899498763, 16267896905895, 117347641620435, 845043416086635, 6076092412278465, 43629213402099045, 312892629725930121, 2241442380182752209

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A163869, A387211.

Cf. A360318, A385813.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^3))

n*a(n) = (10*n-1)*a(n-1) - 21*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).

A387234 Expansion of sqrt((1-3x) / (1-7x)^5).

Original entry on oeis.org

1, 16, 187, 1908, 18015, 161700, 1400385, 11808480, 97533075, 792374720, 6350977457, 50334074972, 395137260609, 3076728075036, 23787996024015, 182783869074000, 1396834725138435, 10622886492055680, 80436297856668225, 606683298398776620, 4559675718517366461

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A387228, A387233.

Cf. A360318, A387212.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^5))

n*a(n) = (10*n+6)*a(n-1) - 21*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).

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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A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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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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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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A383948 Expansion of 1/sqrt((1-3*x)^3 * (1-7*x)).

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A387212 Expansion of sqrt((1-3*x) / (1-7*x)^3).

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A387234 Expansion of sqrt((1-3*x) / (1-7*x)^5).

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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).

A383948 Expansion of 1/sqrt((1-3x)^3 (1-7*x)).

A387212 Expansion of sqrt((1-3x) / (1-7x)^3).

A387234 Expansion of sqrt((1-3x) / (1-7x)^5).