A085364 -id:A085364 - 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)

A085363 a(0)=1, for n>0: a(n) = 49^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

Original entry on oeis.org

1, 4, 28, 212, 1676, 13604, 112380, 940020, 7936620, 67494980, 577309148, 4961187092, 42801458764, 370478720356, 3215827927228, 27982214082612, 244004165618220, 2131710838837380, 18654504783815580, 163488269572628820

Offset: 0

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

Apparently, the number of 2-D directed walks of semilength n starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding adjacent NW/SE and adjacent NE/SW. - David Scambler, Jun 20 2013
Form an array with m(0,n) = m(n,0) = 2^n; m(i,j) equals the sum of the terms to the left of m(i,j) and the sum of the terms above m(i,j), which is m(i,j) = Sum_{k-0..j-1} m(i,k) + Sum_{k=0..i-1} m(k,j).  m(n,n) = a(n). - J. M. Bergot, Jul 10 2013
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=2. (End)
The number of elements in the free group on two generators of length 2n that are zero exponent sum. - Tey Berendschot, Aug 09 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Kees A.S. Immink and Kui Cai, Properties and constructions of constrained codes for DNA-based data storage, IEEE Access, vol. 8, no. 1, pp. 49523-49531, 2020, page 49529.
John Machacek, Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats, preprint arXiv:2105.02417 [math.CO], 2021.

Cf. A001019 (9^n), A084771, A085362, A085364, diagonal of A348595.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-9*x)) )); // G. C. Greubel, May 23 2020

seq(coeff(series(sqrt((1-x)/(1-9*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020

CoefficientList[Series[Sqrt[(1-x)/(1-9x)], {x, 0, 25}], x]

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-9*x)) ) \\ Joerg Arndt, May 10 2013

def A085363_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-9*x)) ).list()
A085363_list(30) # G. C. Greubel, May 23 2020

G.f.: sqrt((1-x)/(1-9*x)).
Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 9^n.
From Vladeta Jovovic, Oct 10 2003: (Start)
First differences of A084771.
a(n) = Sum_{k=1..n} 2^k * binomial(n-1, k-1) * binomial(2*k, k). (End)
D-finite with recurrence n*a(n) = (10*n-6)*a(n-1) - (9*n-18)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) ~ 2*sqrt(2)*3^(2*n-1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
a(0) = 1; a(n) = (4/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} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 12, 74, 466, 2982, 19320, 126390, 833220, 5527190, 36852052, 246751854, 1658106394, 11176100138, 75528743352, 511600414554, 3472363279170, 23609924743590, 160788499672020, 1096566516149790, 7488135911236806, 51193972101241362, 350368409215623192

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360319, A360321, A360322.

Cf. A085364, A122898.

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

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

G.f.: sqrt( (1-3*x)/(1-7*x) ).
n*a(n) = 2*(5*n-4)*a(n-1) - 21*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/3)^i * a(j) * a(i-j) = (7/3)^n.
a(n) = 2 * A122898(n-1) for n > 0.
a(n) ~ 2 * 7^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(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). (End)

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

Original entry on oeis.org

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)

A387230 Expansion of sqrt((1-x) / (1-13*x)^5).

Original entry on oeis.org

1, 32, 723, 14044, 250415, 4224732, 68565049, 1081299296, 16679767923, 252819395920, 3777709472537, 55782986878164, 815526073468561, 11821376147023268, 170096339292264375, 2431786467331116016, 34569517907583692867, 488963045591838160848, 6885041951078984405449

Offset: 0

Seiichi Manyama, Aug 23 2025

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

Cf. A387228, A387229.

Cf. A085364, A387210.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-13*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=20, x='x+O('x^N)); Vec(sqrt((1-x)/(1-13*x)^5))

n*a(n) = (14*n+18)*a(n-1) - 13*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^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^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * n^(3/2) * 13^(n - 1/2) / sqrt(3*Pi). - Vaclav Kotesovec, Aug 24 2025

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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A085363 a(0)=1, for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

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A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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

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A387230 Expansion of sqrt((1-x) / (1-13*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).

A085363 a(0)=1, for n>0: a(n) = 49^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).