A127632 -id:A127632 - cp's OEIS frontend

A130655 Catalan transform of Catalan numbers C(n+1).

1, 2, 7, 28, 119, 524, 2363, 10844, 50446, 237280, 1126437, 5389916, 25967972, 125868952, 613385075, 3003586196, 14771851093, 72936101780, 361419276386, 1796837068400, 8960207761500

Offset: 0

Philippe Deléham, Jun 21 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A127632, A129442.

A130655 := proc(n)
    add(A106566(n,k)*A000108(k+1),k=0..n) ;
end proc: # R. J. Mathar, Mar 01 2015

CoefficientList[Series[2/(Sqrt[-1 + 2*Sqrt[1-4*x]] + Sqrt[1-4*x]),{x,0,20}],x] (* Vaclav Kotesovec, Jul 02 2015 *)

x='x+O('x^50); Vec(2/(sqrt(-1 + 2*sqrt(1-4*x)) + sqrt(1-4*x))) \\ G. C. Greubel, Mar 21 2017

a(n) = Sum_{k=0..n} A106566(n,k)*A000108(k+1).
Conjecture: 3*n*(n-2)*(n+2)*a(n) +4*(-10*n^3+21*n^2+7*n-15)*a(n-1) +16*(11*n^3-47*n^2+57*n-15)*a(n-2) -8*(2*n-5)*(4*n-9)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Mar 01 2015
G.f.: (C(x*C(x))-1)/(x*C(x)), where C(x) is g.f. of Catalan numbers A000108. - Vladimir Kruchinin, Jul 02 2015
a(n) ~ 2^(4*n+3/2) / (sqrt(Pi) * n^(3/2) * 3^(n-1/2)). - Vaclav Kotesovec, Jul 02 2015

A344056 a(n) = (2n)! hypergeom([3/2, 2, 1-n], [3, 2 - 2*n], 4) for n >= 1 and a(0) = 1.

Original entry on oeis.org

1, 2, 72, 3960, 354816, 47952000, 9169459200, 2363720486400, 791505727488000, 334400903553024000, 174128130895380480000, 109627479651269099520000, 82122139255681984757760000, 72210265570358144925696000000, 73668698593180294904178278400000, 86324332785379145107697958912000000

Offset: 0

Peter Luschny, May 12 2021

nonn

Cf. A127632.

a[n_] := (2 n)! HypergeometricPFQ[{3/2, 2, 1 - n}, {3, 2 - 2n}, 4]; a[0] := 1;
Table[a[n], {n, 0, 15}]

a(n) * Catalan(n - 1) = (2*n)! * A127632(n) for n >= 1.

A350645 Number of permutations avoiding 132 of length 3n composed of only 3-cycles.

Original entry on oeis.org

1, 2, 8, 36, 170, 824, 4060, 20232, 101664, 514140, 2613468, 13340496, 68335644, 351087128, 1808405600, 9335697424, 48289295226, 250213951992, 1298517484804, 6748250144600, 35114221973600, 182924946400680, 953931045159000, 4979398271047200, 26014703727203100

Offset: 0

Kassie Archer, Jan 09 2022

nonn

Also the number of permutations avoiding 213 of length 3n composed of only 3-cycles.

			For n=2, the eight permutations (in one-line notation and cycle notation) are:
  [6, 5, 2, 1, 3, 4] (1,6,4)(2,5,3)
  [6, 4, 2, 3, 1, 5] (1,6,5)(2,4,3)
  [6, 3, 4, 2, 1, 5] (1,6,5)(2,3,4)
  [5, 6, 1, 2, 3, 4] (1,5,3)(2,6,4)
  [3, 4, 5, 6, 1, 2] (1,3,5)(2,4,6)
  [4, 3, 5, 6, 2, 1] (1,4,6)(2,3,5)
  [5, 3, 4, 2, 6, 1] (1,5,6)(2,3,4)
  [5, 4, 2, 3, 6, 1] (1,5,6)(2,4,3) .

Kassie Archer and Christina Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.

Cf. A000108, A001006, A127632.

G.f.: c(x*c(x))/(2-c(x*c(x))) where c(x) is the generating function for Catalan numbers. Notice c(x*c(x)) is given in A127632.

G.f.: (1+A(x))/(1-A(x)) where A(x) = (c(x)-1)*c(m(x)-1) where c(x) is the generating function for Catalan numbers and m(x) is the generating function for the Motzkin numbers.

A366114 Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^3 ).

Original entry on oeis.org

1, 2, 4, 7, 9, 2, -34, -130, -284, -284, 730, 4864, 14860, 27134, 6462, -170865, -771303, -2005828, -2751028, 3491747, 36288137, 130265102, 283131062, 210905402, -1317613954, -7461822262, -22297519418, -38398674146, 10151248222, 355843715494, 1495838414326

Offset: 0

Seiichi Manyama, Sep 29 2023

sign

Cf. A127632, A364374, A366115.

Cf. A279565.

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

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

A381861 G.f. A(x) satisfies A(x) = (1 + xA(x))^4 C(x), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 5, 32, 231, 1797, 14715, 125064, 1093194, 9766783, 88793815, 818832674, 7640868924, 72014955566, 684551660324, 6555290711728, 63179148757584, 612376024087047, 5965515657187437, 58375460484257734, 573545171374958628, 5655759227878768987, 55957005428512022905

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A000108, A127632, A153299.

Cf. A381877.

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

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

a(n) = binomial(4 + 4*n, n)*hypergeom([-4/3-n, -2/3-n, -n, 1+n], [-3/4-n, -1/2-n, -1/4-n], 3^3/2^8)/(1 + n). - Stefano Spezia, Mar 09 2025

A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

Original entry on oeis.org

1, -1, 2, -3, 2, 4, -18, 43, -80, 123, -148, 78, 287, -1364, 3858, -8627, 15901, -23076, 20061, 18294, -140623, 420241, -930040, 1655753, -2293975, 1872682, 1835066, -12983537, 37871888, -83222132, 149287250, -212064236, 186932259, 131172644, -1139053896, 3449157957, -7710640256

Offset: 0

Ilya Gutkovskiy, Nov 29 2017

sign

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A003823, A007325, A055101, A055102, A055103, A127632, A286509, A291651.

nmax = 36; CoefficientList[Series[1/(1 + ContinuedFractionK[(x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
g[x_] := g[x] = QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]); a[n_] := a[n] = SeriesCoefficient[g[x g[x]], {x, 0, n}];  Table[a[n], {n, 0, 36}]

G.f.: 1/(1 + x/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))/(1 + x^2/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^2/(1 + x^3/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^3/(1 + ...)))), a continued fraction.

cp's OEIS Frontend

A130655 Catalan transform of Catalan numbers C(n+1).

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A344056 a(n) = (2*n)! * hypergeom([3/2, 2, 1-n], [3, 2 - 2*n], 4) for n >= 1 and a(0) = 1.

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A350645 Number of permutations avoiding 132 of length 3n composed of only 3-cycles.

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A366114 Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^3 ).

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A381861 G.f. A(x) satisfies A(x) = (1 + x*A(x))^4 * C(x), where C(x) is the g.f. of A000108.

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A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

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Crossrefs

Programs

Mathematica

Formula

A344056 a(n) = (2n)! hypergeom([3/2, 2, 1-n], [3, 2 - 2*n], 4) for n >= 1 and a(0) = 1.

A381861 G.f. A(x) satisfies A(x) = (1 + xA(x))^4 C(x), where C(x) is the g.f. of A000108.