A025230 -id:A025230 - cp's OEIS frontend

A068769 Generalized Catalan numbers 7xA(x)^2 -A(x) +1 -6*x=0.

1, 1, 14, 203, 3038, 46746, 736764, 11853051, 194053622, 3224557406, 54265836548, 923218762270, 15854602773100, 274500192707860, 4786546243533432, 83989334625037947, 1481965556616225702

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(7,7; n)/7 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..780

Cf. A000108, A068764-A068768, A068770-A068772, A025227-A025230.

a[0] = 1; a[1] = 1; a[2] = 14; a[n_] := (168 (2 - n) a[n - 2] + 14 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-28*x*(1-6*x)])/(14*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (7^n) * p(n, -6/7) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 7*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-28*x*(1-6*x)))/(14*x).
Recurrence: (n+1)*a(n) = 168*(2-n)*a(n-2) + 14*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(14+14*sqrt(7)) * (14+2*sqrt(7))^n / (14*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A068770 Generalized Catalan numbers 8xA(x)^2 -A(x) +1 -7*x=0.

Original entry on oeis.org

1, 1, 16, 264, 4480, 77952, 1386496, 25135616, 463233024, 8658673664, 163829383168, 3132565553152, 60446638866432, 1175715287400448, 23028562592268288, 453848132868898816, 8993594212565909504

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(8,8; n)/8 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..750

Cf. A000108, A068764-A068769, A068771-A068772, A025227-A025230.

CoefficientList[Series[(1-Sqrt[1-32*x*(1-7*x)])/(16*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = if(n, (4^(n-1)*14^(1/2*n+1/2)*pollegendre(n+1,2/7*14^(1/2)) - pollegendre(n,2/7*14^(1/2))*4^n*14^(n/2))\/n, 1) \\ Charles R Greathouse IV, Mar 19 2017

a(n) = (8^n) * p(n, -7/8) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 8*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-32*x*(1-7*x)))/(16*x).
a(n) = (4^(n-1)*14^(1/2*n+1/2)*LegendreP(n+1,2/7*14^(1/2)) - LegendreP(n,2/7*14^(1/2))*4^n*14^(1/2*n))/n for n > 0. - Mark van Hoeij, Apr 23 2010
Recurrence: (n+1)*a(n) = 224*(2-n)*a(n-2) + 16*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(1+2*sqrt(2)) * (16+4*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

Original entry on oeis.org

1, 1, 18, 333, 6318, 122634, 2429028, 48974949, 1002875094, 20814628158, 437088964860, 9272342710962, 198456435657036, 4280758166952756, 92972201833888200, 2031520673763657621, 44630859892110807654

Offset: 0

Wolfdieter Lang, Mar 04 2002

Fung Lam, Table of n, a(n) for n = 0..725

Cf. A000108, A068764-A068770, A068772, A025227-A025230.

a[n_] := (288 (2 - n) a[n - 2] + 18 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}](* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-36*x*(1-8*x)])/(18*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (9^n) * p(n, -8/9) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 9*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-36*x*(1-8*x)))/(18*x).
Recurrence: (n+1)*a(n) = 288*(2-n)*a(n-2) + 18*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(2) * 24^n / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 4, 2, 16, 16, 5, 64, 96, 60, 14, 256, 512, 480, 224, 42, 1024, 2560, 3200, 2240, 840, 132, 4096, 12288, 19200, 17920, 10080, 3168, 429, 16384, 57344, 107520, 125440, 94080, 44352, 12012, 1430, 65536, 262144, 573440, 802816, 752640, 473088, 192192, 45760, 4862

Offset: 0

Peter Luschny, Feb 03 2015

			[   1]
[   4,     2]
[  16,    16,     5]
[  64,    96,    60,    14]
[ 256,   512,   480,   224,    42]
[1024,  2560,  3200,  2240,   840,  132]
[4096, 12288, 19200, 17920, 10080, 3168, 429]

Cf. A108198 (Peter Bala), A000302, A000108, A025230, A002699, A128650, A254633.

h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);

T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019 *)

A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)
for n in range(7): [A254632(n,k) for k in (0..n)]

T(n,0) = A000302(n).
T(n,n) = A000108(n+1).
T(n,1) = A002699(n) for n>=1.
T(n,n-1) = A128650(n+2) for n>=1.
T(2*n,n) = A254633(n).
T(n,k) = 4^(n-k)*C(n,k)*Catalan(k+1).
sum(k=0..n, T(n,k)) = A025230(n+2).

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

A349532 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 4 * x).

Original entry on oeis.org

1, 1, 7, 52, 407, 3329, 28232, 246552, 2204895, 20103027, 186223399, 1748009560, 16591329652, 158975004204, 1535725632552, 14940742412112, 146259921123407, 1439658075118967, 14240062489572485, 141469058343614452, 1410975387252602527, 14122900638031585153

Offset: 0

Ilya Gutkovskiy, Nov 21 2021

nonn

In general, if k >= 0 and g.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - k*x), then a(n) ~ (4*k + 27)^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Nov 25 2021

Cf. A001764, A025230, A307678, A349318, A349531.

nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x]^3/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]

a(0) = a(1) = 1; a(n) = 4 * a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(3*k,k) * 4^(n-k) / (2*k+1).
a(n) = 4^(n-1)*F([4/3, 5/3, 1-n], [2, 5/2], -3^3/2^4), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021
a(n) ~ 43^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Nov 25 2021

A218186 Number of rows with the value true in the truth tables of all bracketed formulas with n distinct propositions p_1, ..., p_n connected by the binary connective of m-implication (case 1).

Original entry on oeis.org

0, 0, 1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, 22206420, 157027938, 1120292388, 8055001716, 58314533400, 424740506109, 3110401363122, 22888001498102, 169155516667524, 1255072594261142, 9345400450314924, 69812926066668044, 523072984217339304, 3929809142578361938, 29598511892723647860

Offset: 0

N. J. A. Sloane, Oct 23 2012

nonn

Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

Essentially the same as A025230.

my(x='x+O('x^30)); concat([0,0], Vec((1-6*x-sqrt((1-4*x)*(1-8*x)))/2)) \\ Michel Marcus, Oct 21 2020

Yildiz gives a g.f.
G.f.: (1-6*x-sqrt((1-4*x)*(1-8*x)))/2. - Michel Marcus, Oct 21 2020
D-finite with recurrence n*a(n) +(n+5)*a(n-1) +(n+44)*a(n-2) +(n+331)*a(n-3) +10*(-902*n+3859)*a(n-4) +34720*(n-6)*a(n-5)=0. - R. J. Mathar, Nov 22 2023

a(6) corrected by Georg Fischer, Jun 07 2021

cp's OEIS Frontend

A068769 Generalized Catalan numbers 7*x*A(x)^2 -A(x) +1 -6*x=0.

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A068770 Generalized Catalan numbers 8*x*A(x)^2 -A(x) +1 -7*x=0.

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A068771 Generalized Catalan numbers 9*x*A(x)^2 -A(x) +1 -8*x=0.

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A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

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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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A349532 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 4 * x).

A068769 Generalized Catalan numbers 7xA(x)^2 -A(x) +1 -6*x=0.

A068770 Generalized Catalan numbers 8xA(x)^2 -A(x) +1 -7*x=0.

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

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).