A025227 -id:A025227 - cp's OEIS frontend

A366694 G.f. satisfies A(x) = (1 + x)^2 + x*A(x)^2.

1, 3, 7, 23, 88, 363, 1576, 7091, 32768, 154588, 741442, 3604495, 17721394, 87960004, 440165522, 2218289051, 11248850578, 57354875692, 293860786178, 1512169500356, 7811933144432, 40499933496818, 210643657689644, 1098802033533295, 5747266778089846

Offset: 0

Seiichi Manyama, Oct 16 2023

nonn

Cf. A025227, A366695.

Cf. A086616.

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

G.f.: A(x) = 2*(1+x)^2 / (1+sqrt(1-4*x*(1+x)^2)).

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

A366695 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^2.

Original entry on oeis.org

1, 4, 11, 39, 166, 765, 3716, 18725, 96956, 512690, 2756806, 15027651, 82853678, 461215414, 2588619402, 14632777719, 83232244238, 476040155118, 2736005962314, 15793863291792, 91530881427964, 532343678619778, 3106141476531628, 18177446846299299

Offset: 0

Seiichi Manyama, Oct 16 2023

nonn

Cf. A025227, A366694.

Cf. A162481.

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

G.f.: A(x) = 2*(1+x)^3 / (1+sqrt(1-4*x*(1+x)^3)).

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

A379171 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^3.

Original entry on oeis.org

1, 4, 21, 139, 1021, 8010, 65708, 556751, 4834686, 42800265, 384832083, 3504693519, 32261240127, 299685628629, 2805773759322, 26448278629697, 250806022116194, 2390973659474304, 22901157688878983, 220279614235505630, 2126890041331033797, 20606993367985131716

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A365764, A369215, A379172.

Cf. A025227, A379173.

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

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

A103970 Expansion of (1 - sqrt(1 - 4x - 12x^2))/(2*x).

Original entry on oeis.org

1, 4, 8, 32, 128, 576, 2688, 13056, 65024, 330752, 1710080, 8962048, 47497216, 254132224, 1370849280, 7447117824, 40707293184, 223731253248, 1235630948352, 6853893292032, 38166664839168, 213288826699776, 1195775593807872, 6723691157127168, 37908469021409280, 214260335517892608, 1213784937073737728, 6890689428042285056

Offset: 0

Paul Barry, Feb 23 2005

Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x) -> (1+3x)g(x(1+3x)). In general, the image of the Catalan numbers under the mapping g(x) -> (1+i*x)g(x(1+i*x)) is given by a(n) = Sum_{k=0..n} i^(n-k)*C(k)*C(k+1,n-k).
Hankel transform is 4^C(n+1,2)*A128018(n). [Paul Barry, Nov 20 2009]
By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we also obtain (n+1)*C(n) - 2*a*(2*n-1)*C(n-1) + 4*(n-2)*(a^2-b)*C(n-2) = 0. In the present case, we also have the asymptotic result: a(n) ~ sqrt(4/3)*2^(n-1)*3^(n+1)/sqrt(Pi*n^3) for large n. - Richard Choulet, Dec 17 2009

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

Cf. A025227, A025229, A103971, A103972.

Cf. A000108, A025228, A025230, A025231, A025232. [Richard Choulet, Dec 17 2009]

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( (1-Sqrt(1-4*x-12*x^2))/(2*x) )); // G. C. Greubel, Mar 16 2019

n:=30:a(0):=1:a(1):=4: k:=1: for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)','p'=0..k):od:seq(a(k),k=0..n); # Richard Choulet, Dec 17 2009
taylor(((1-(1-4*z-12*z^2)^0.5)/(2*z)),z=0,32); # Richard Choulet, Dec 17 2009

CoefficientList[Series[(1 - Sqrt[1-4x-12x^2])/(2x), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 18 2017 *)

my(x='x+O('x^35)); Vec((1-sqrt(1-4*x-12*x^2))/(2*x)) \\ G. C. Greubel, Mar 16 2019

((1-sqrt(1-4*x-12*x^2))/(2*x)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Mar 16 2019

G.f.: (1 - sqrt(1-4*x*(1+3*x)))/(2*x).
a(n) = Sum_{k=0..n} 3^(n-k)*C(k)*C(k+1, n-k).
D-finite with recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + 12*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009

A103971 Expansion of (1 - sqrt(1 - 4x - 16x^2))/(2*x).

Original entry on oeis.org

1, 5, 10, 45, 190, 930, 4660, 24445, 131190, 719830, 4013260, 22684370, 129661740, 748252580, 4353379560, 25508284445, 150392391590, 891549228430, 5310994644060, 31775749689670, 190860711108740, 1150473009844380

Offset: 0

Paul Barry, Feb 23 2005

Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x) -> (1+4x)g(x(1+4x)). In general, the image of the Catalan numbers under the mapping g(x)->(1+i*x)g(x(1+i*x)) is given by a(n) = Sum_{k=0..n} i^(n-k)C(k)C(k+1,n-k).

More generally, the sequence C for which C(0)=a, C(1)=b and C(n+1) = sum(C(k)*C(n-k),k=0..n) has the following g.f. f: f(z) = (1-sqrt(1-4*z*(a-(a^2-b)*z)))/(2*z). We obtain: C(n)=(sum(-1)^(p-1)*2^{n-p}a^{n-2*p-1}*(a^2-b)^p*((2*n-2*p-1)*...*5*3*1/(p!*(n-2*p+1)!)),p=0..floor((n+1)/2)). By following Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we obtain also: (n+1)*C(n) - 2*a*(2*n-1)*C(n-1) + 4*(n-2)*(a^2-b)*C(n-2) = 0. - Richard Choulet, Dec 17 2009

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

Cf. A000108, A025227, A025228, A025229, A025230, A025231, A025232. - Richard Choulet, Dec 17 2009

n:=30:a(0):=1:a(1):=5: for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)','p'=0..k):od:seq(a(k),k=0..n); # Richard Choulet, Dec 17 2009

CoefficientList[Series[(1-Sqrt[1-4x-16x^2])/(2x),{x,0,30}],x] (* Harvey P. Dale, Apr 02 2012 *)

G.f.: (1-sqrt(1-4*x*(1+4*x)))/(2*x).
a(n) = Sum_{k=0..n} 4^(n-k)*C(k)*C(k+1, n-k).
Another recurrence formula: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + 16*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009
a(n) ~ sqrt(10 + 2*sqrt(5))*(2 + 2*sqrt(5))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
Equivalently, a(n) ~ 5^(1/4) * 2^(2*n) * phi^(n + 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

A103972 Expansion of (1-sqrt(1-4x-20x^2))/(2*x).

Original entry on oeis.org

1, 6, 12, 60, 264, 1392, 7392, 41424, 236640, 1384512, 8224896, 49554816, 301884672, 1856878080, 11514915840, 71915838720, 451938731520, 2855705994240, 18132621772800, 115637702461440, 740356410961920, 4756888756101120, 30662391191715840, 198229520200704000, 1285001080928845824

Offset: 0

Paul Barry, Feb 23 2005

Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x)->(1+5x)g(x(1+5x)). In general, the image of the Catalan numbers under the mapping g(x)->(1+i*x)g(x(1+i*x)) is given by a(n)=sum{k=0..n, i^(n-k)C(k)C(k+1,n-k)}.

More generally, the sequence C for which C(0)=a, C(1)=b and C(n+1)=sum(C(k)*C(n-k),k=0..n) has the following G.f f: f(z)= (1-sqrt(1-4*z*(a-(a^2-b)*z)))/(2*z). We obtain: C(n)=(sum(-1)^(p-1)*2^{n-p}a^{n-2*p-1}*(a^2-b)^p*((2*n-2*p-1)*...*5*3*1/(p!*(n-2*p+1)!)),p=0..floor((n+1)/2)). By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we obtain also: (n+1)*C(n)-2*a*(2*n-1)*C(n-1)+4*(n-2)*(a^2-b)*C(n-2)=0. - Richard Choulet, Dec 17 2009

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

Cf. A000108, A025227, A025228, A025229, A025230, A025231, A025232. [From Richard Choulet, Dec 17 2009]

n:=30:a(0):=1:a(1):=6 :for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)','p'=0..k):od:seq(a(k),k=0..n); # Richard Choulet, Dec 17 2009

CoefficientList[Series[(1-Sqrt[1-4*x-20*x^2])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec((1-sqrt(1-4*x-20*x^2))/(2*x)) \\ Joerg Arndt, May 13 2013

G.f.: (1-sqrt(1-4*x*(1+5*x)))/(2*x).
a(n) = Sum_{k=0..n} 5^(n-k)*C(k)*C(k+1, n-k).
Another recurrence formula: (n+1)*a(n)=2*(2n-1)*a(n-1)+20*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009
a(n) ~ sqrt(12+2*sqrt(6))*(2+2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

A260772 Certain directed lattice paths.

Original entry on oeis.org

1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.

Cf. A122868, A260774, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

# A260772 satisfies a 4th-order recurrence that can be reduced
# to a 2nd-order recurrence given in this program t:
t := proc(n) options remember;
if n <= 1 then
    [-1/2, 0, 1, 4][2*n+2]
  else
    (16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1))
       /( n*(2*n+1)*(5*n-7) )
  fi
end:
A260772 := proc(n)
t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n)
end:
seq(A260772(i),i=0..100);
# Mark van Hoeij, Jul 14 2022

a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */

a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019

G.f.: P1(x) = (2*(1-x)/3)/x - ((2*sqrt(1-5*x-2*x^2)/3)/x)*sin((Pi/6 + arccos(((20*x^3-6*x^2+15*x-2)/2)/(1-5*x-2*x^2)^(3/2))/3)). - See Dziemianczuk (2014), Proposition 11.
a(n) = (1/n)*Sum_{j=0..(n+1)/4} (-1)^j*C(n,j)*C(3*n-4*j,n-4*j+1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
n*(n+1)*(25*n^2-70*n+21)*a(n) - 30*(7*n-15)*n*a(n-1) + (-1100*n^4+5280*n^3-6424*n^2-1188*n+3816)*a(n-2) + 120*(n+2)*(n-3)*a(n-3) - 16*(n-3)*(n-4)*(25*n^2-20*n-24)*a(n-4) = 0. - Mark van Hoeij, Jul 14 2022
a(n) ~ 2^(n - 1/2) * phi^((10*n - 1)/4) / (sqrt(Pi) * 5^(1/4) * sqrt(phi^(3/2) - 2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 15 2022

More terms from Lars Blomberg, Aug 01 2015

A367042 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^2.

Original entry on oeis.org

1, 1, 2, 6, 16, 48, 152, 500, 1688, 5816, 20368, 72288, 259424, 939808, 3432192, 12622416, 46706144, 173762016, 649569216, 2438748864, 9191656192, 34765298944, 131912452864, 501987944832, 1915417307392, 7326620001536, 28088736525824, 107913607531520

Offset: 0

Seiichi Manyama, Nov 03 2023

nonn

Cf. A366676, A367043.
Cf. A025227, A176697.
Cf. A226022.

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

G.f.: A(x) = 2*(1+x^3) / (1+sqrt(1-4*x*(1+x^3))).

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

A367639 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^2 / (1 + x).

Original entry on oeis.org

1, 3, 6, 16, 52, 184, 688, 2672, 10672, 43552, 180800, 761088, 3241088, 13937408, 60435968, 263962880, 1160188672, 5127762432, 22775636992, 101608357888, 455105255424, 2045751037952, 9225923895296, 41731062358016, 189275050729472, 860630181167104

Offset: 0

Seiichi Manyama, Nov 25 2023

Cf. A006318, A025227, A367640, A333090, A367641.

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

G.f.: A(x) = 2*(1+x)^2 / (1+sqrt(1-4*x*(1+x))).
a(n) = Sum_{k=0..n} binomial(k+2,n-k) * binomial(2*k,k)/(k+1).
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 25 2023
D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-4*n+9)*a(n-2) +4*(-n+4)*a(n-3)=0. - R. J. Mathar, Dec 04 2023
From Peter Bala, May 05 2024: (Start)
A(x) = (1 + x)*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318. Cf. A025227.
A333090(n) = [x^n] A(x)^n. (End)

A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2r+k,n)/(2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 12, 0, 1, 8, 24, 40, 40, 0, 1, 10, 40, 92, 144, 144, 0, 1, 12, 60, 176, 360, 544, 544, 0, 1, 14, 84, 300, 752, 1440, 2128, 2128, 0, 1, 16, 112, 472, 1400, 3200, 5872, 8544, 8544, 0, 1, 18, 144, 700, 2400, 6352, 13664, 24336, 35008, 35008, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   2,    4,    6,     8,    10,    12, ...
  0,   4,   12,   24,    40,    60,    84, ...
  0,  12,   40,   92,   176,   300,   472, ...
  0,  40,  144,  360,   752,  1400,  2400, ...
  0, 144,  544, 1440,  3200,  6352, 11616, ...
  0, 544, 2128, 5872, 13664, 28480, 54768, ...

Columns k=0..1 give A000007, A025227(n+1).
Main diagonal gives A333473.
Cf. A266213, A378318.

T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(2/k) )^k for k > 0.
G.f. of column k: (B(x)/x)^k where B(x) is the g.f. of A025227.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+1) for n > 0.

cp's OEIS Frontend

A366694 G.f. satisfies A(x) = (1 + x)^2 + x*A(x)^2.

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A366695 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^2.

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A379171 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^3.

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A103970 Expansion of (1 - sqrt(1 - 4*x - 12*x^2))/(2*x).

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A103971 Expansion of (1 - sqrt(1 - 4*x - 16*x^2))/(2*x).

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A103972 Expansion of (1-sqrt(1-4*x-20*x^2))/(2*x).

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A260772 Certain directed lattice paths.

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A367042 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^2.

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A103970 Expansion of (1 - sqrt(1 - 4x - 12x^2))/(2*x).

A103971 Expansion of (1 - sqrt(1 - 4x - 16x^2))/(2*x).

A103972 Expansion of (1-sqrt(1-4x-20x^2))/(2*x).

A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2r+k,n)/(2r+k) for k > 0.