A085363 -id:A085363 - 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)

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

Original entry on oeis.org

1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725

Offset: 0

Paul D. Hanna, Jun 10 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Sums of squares of coefficients of (1+2*x)^n. - Joerg Arndt, Jul 06 2011
The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007
Partial sums of A085363. - J. M. Bergot, Jun 12 2013
Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Tewodros Amdeberhan, In search of multiple expressions for a sequence
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], 2015.
Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See p. 3.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Cf. A001850, A059231, A059304, A246923 (a(n)^2).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # Muniru A Asiru, Jul 29 2018

[3^n*Evaluate(LegendrePolynomial(n), 5/3) : n in [0..40]]; // G. C. Greubel, May 30 2023

seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # Peter Luschny, Apr 26 2016

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)
Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* Vladimir Reshetnikov, Nov 29 2013 *)
CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* Harvey P. Dale, Mar 08 2016 *)
Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)

{a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* Michael Somos, Oct 08 2003 */

{a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* Michael Somos, Oct 08 2003 */

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1], [1,1]]; /* note the triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

[3^n*gen_legendre_P(n, 0, 5/3) for n in range(41)] # G. C. Greubel, May 30 2023

G.f.: 1 / sqrt(1 - 10*x + 9*x^2).
From Vladeta Jovovic, Aug 20 2003: (Start)
Binomial transform of A059304.
G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).
E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)
a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - Paul Barry, May 19 2006
a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 11 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - R. J. Mathar, Nov 26 2012
a(n) = hypergeom([-n, -n], [1], 4). - Vladimir Reshetnikov, Nov 29 2013
a(n) = hypergeom([-n, 1/2], [1], -8). - Peter Luschny, Apr 26 2016
From Michael Somos, Jun 01 2017: (Start)
a(n) = -3 * 9^n * a(-1-n) for all n in Z.
0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)
From Peter Bala, Nov 13 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)
a(n) = 3^n * LegendreP(n, 5/3). - G. C. Greubel, May 30 2023
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

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)

A085364 a(0)=1, for n>0: a(n) = 613^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

Original entry on oeis.org

1, 6, 60, 654, 7458, 87378, 1042152, 12587730, 153479508, 1885010946, 23285957604, 289018502682, 3601315495050, 45023019250398, 564465885846216, 7094214579174558, 89351097367355826, 1127492973620753010

Offset: 0

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

From G. C. Greubel, May 23 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).
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
(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=3. (End)

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

Cf. A001022 (13^n), A085362, A085363.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-13*x)) )); // G. C. Greubel, May 23 2020

seq(coeff(series( sqrt((1-x)/(1-13*x)) , x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020

CoefficientList[Series[Sqrt[(1-x)/(1-13x)], {x, 0, 25}], x]

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-13*x))) \\ Joerg Arndt, May 10 2013

def A085362_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-13*x)) ).list()
A085362_list(30) # G. C. Greubel, May 23 2020

G.f.: sqrt((1-x)/(1-13*x))
Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 13^n.
D-finite with recurrence: n*a(n) = 2*(7*n-4)*a(n-1) - 13*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 2*sqrt(3)*13^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
a(0) = 1; a(n) = (6/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} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(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). (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)

A387208 Expansion of sqrt((1-x) / (1-9*x)^3).

Original entry on oeis.org

1, 13, 145, 1517, 15329, 151565, 1476465, 14228205, 135990465, 1291409165, 12199991633, 114761111789, 1075651464865, 10051341904141, 93677905064497, 871083359663085, 8083754402585985, 74885500462111245, 692624008942816785, 6397104350057979885, 59008673876627412321

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A163869, A163872, A387210.

Cf. A085363, A383946.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1-x) / (1-9*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-9*x)^3))

n*a(n) = (10*n+3)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 2^(5/2) * sqrt(n) * 3^(2*n-1) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025

A387229 Expansion of sqrt((1-x) / (1-9*x)^5).

Original entry on oeis.org

1, 22, 343, 4604, 56765, 662450, 7438515, 81174840, 866564025, 9090485390, 94014360143, 960890353076, 9723664642549, 97564323687082, 971756818248235, 9616894723897200, 94635806917660785, 926607762721058310, 9032093873432341575, 87685949210949054060, 848182216775168898861

Offset: 0

Seiichi Manyama, Aug 23 2025

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

Cf. A387228, A387230.

Cf. A085363, A387208.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-9*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

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

n*a(n) = (10*n+12)*a(n-1) - 9*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^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} 2^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,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).
a(n) ~ 2^(7/2) * n^(3/2) * 3^(2*n-2) / sqrt(Pi). - Vaclav Kotesovec, Aug 24 2025

A348595 Triangle read by rows: Number of walks from (0,0) to (3n,3k) on the square lattice with up and right steps where squares (x,y)=(1,1) mod 3 or (x,y)=(2,2) mod 3 are not entered.

Original entry on oeis.org

1, 1, 4, 1, 8, 28, 1, 12, 64, 212, 1, 16, 116, 520, 1676, 1, 20, 184, 1052, 4288, 13604, 1, 24, 268, 1872, 9316, 35784, 112380, 1, 28, 368, 3044, 17976, 81708, 301440, 940020, 1, 32, 484, 4632, 31740, 167376, 713940, 2558280, 7936620, 1, 36, 616, 6700, 52336, 314932, 1531000, 6231100, 21842560, 67494980

Offset: 0

R. J. Mathar, Jan 26 2022

			The array is symmetric; the non-redundant triangular part starts
   1
   1    4
   1    8   28
   1   12   64  212
   1   16  116  520  1676
   1   20  184 1052  4288  13604
   1   24  268 1872  9316  35784 112380
   1   28  368 3044 17976  81708 301440  940020
   1   32  484 4632 31740 167376 713940 2558280 7936620

R. J. Mathar, Walks of up and right steps in the square lattice with blocked squares

Cf. A085363 (diagonal), A307584 (walks to (3n+1,3k))

A348595 := proc(n,k)
    g := (1-u*v)/(1-u-v-3*u*v) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,k) ;
end proc:
seq(seq( A348595(n,k),k=0..n),n=0..10) ;

T[n_, k_] := Module[{u, v}, SeriesCoefficient[(1 - u v)/(1 - u - v - 3 u v), {u, 0, n}] // SeriesCoefficient[#, {v, 0, k}]&];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

G.f.: (1-u*v)/(1-u-v-3*u*v) .

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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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A084771 Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.

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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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A085364 a(0)=1, for n>0: a(n) = 6*13^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

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

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A387208 Expansion of sqrt((1-x) / (1-9*x)^3).

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A387229 Expansion of sqrt((1-x) / (1-9*x)^5).

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

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

A085364 a(0)=1, for n>0: a(n) = 613^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).