A052854 -id:A052854 - cp's OEIS frontend

A304787 Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).

1, 1, 2, 7, 20, 67, 222, 758, 2617, 9189, 32554, 116494, 420046, 1525221, 5571065, 20457808, 75476447, 279636977, 1039965746, 3880891892, 14527657602, 54537434161, 205270200229, 774460385687, 2928429307876, 11095878177649, 42122749335654, 160192845018335, 610224764470011

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A052805, A052854, A088327, A292668.

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^CatalanNumber[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d CatalanNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} (1 + x^k)^A000108(k).

a(n) ~ c * 4^n / n^(3/2), where c = exp( Sum_{k>=1} (-1)^k * (2 - 4^k + 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.4863036894111457491052224706533674748514957... - Vaclav Kotesovec, Mar 21 2021

A052805 If B is a collection in which there are C(n-1) [Catalan numbers, A000108] things with n points, a(n) is the number of subsets without repetition of B with a total of n points.

Original entry on oeis.org

1, 1, 1, 3, 7, 21, 64, 204, 666, 2236, 7625, 26419, 92644, 328370, 1174234, 4231898, 15354424, 56042372, 205626906, 758021598, 2806143522, 10427671924, 38882984840, 145443260702, 545598228056, 2052086677666, 7736986142773, 29236241424977

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [1,0,2,4,14,40,132,424,1430,...] (C(n-1) if n odd, C(n-1)-C(n/2-1) if n even).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 766

Cf. A000108, A052854.

spec := [S,{C=Sequence(B),B=Prod(C,Z),S=PowerSet(B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

ClearAll[a]; b[k_] := Sum[ (-1)^(k/d + 1)*Binomial[2*d - 2, d - 1], {d, Divisors[k]}]; a[0] = 1; a[n_] := a[n] = (1/n)*Sum[a[n - k]*b[k], {k, 1, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 08 2012, after Vladeta Jovovic *)

a(n)=local(A); if(n<1,!n,A=sum(k=1,n,(2*k-2)!/k!/(k-1)!*x^k,x*O(x^n)); polcoeff(exp(sum(k=1,n,-(-1)^k*subst(A,x,x^k)/k)),n))

a(n)=(1/n)*Sum_{k=1..n} a(n-k)*b(k), n>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d+1)*binomial(2*d-2, d-1). - Vladeta Jovovic, Jan 17 2002
G.f. A(x)=exp(Sum_{k>0} -(-1)^k* C(x^k)/k) where C(x)=(1-sqrt(1-4x))/2= g.f. A000108 (offset 1).
G.f.: Product_{k>=1} (1+x^k)^(1/k*binomial(2*k-2, k-1)). - Vladeta Jovovic, Jan 17 2002

A246949 Decimal expansion of the coefficient K appearing in the asymptotic expression of the number of forests of ordered trees on n total nodes as K4^(n-1)/sqrt(Pin^3).

Original entry on oeis.org

1, 7, 1, 6, 0, 3, 0, 5, 3, 4, 9, 2, 2, 2, 8, 1, 9, 6, 4, 0, 4, 7, 4, 6, 4, 3, 9, 9, 0, 4, 2, 2, 1, 2, 0, 9, 1, 9, 6, 9, 7, 6, 7, 8, 3, 7, 3, 1, 7, 8, 6, 3, 4, 6, 3, 1, 8, 6, 8, 1, 9, 4, 0, 7, 1, 4, 5, 1, 4, 9, 6, 2, 1, 3, 2, 6, 0, 2, 0, 1, 6, 9, 3, 6, 6, 4, 2, 7, 2, 3, 8, 1, 5, 2, 6, 4, 6, 1, 1, 7, 3, 0, 1, 1, 5

Offset: 1

Jean-François Alcover, Sep 08 2014

See A052854.

			1.7160305349222819640474643990422120919697678373178634631868194...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 45.
Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, and N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.

Cf. A052854.

evalf(exp(sum(1/(2*k)*(1-sqrt(1-4^(1-k))),k=1..infinity)),100); # Vaclav Kotesovec, Sep 17 2014

digits = 76; K = Exp[NSum[1/(2 k)*(1 - Sqrt[1 - 4^(1 - k)]), {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 100]]; RealDigits[K, 10, digits] // First

Equals exp(Sum_{k>=1} (1 - sqrt(1 - 4^(1 - k)))/(2*k)).

More terms from Vaclav Kotesovec, Sep 17 2014

A066768 Sum_{d|n} binomial(2*d-2,d-1).

Original entry on oeis.org

1, 3, 7, 23, 71, 261, 925, 3455, 12877, 48693, 184757, 705713, 2704157, 10401527, 40116677, 155120975, 601080391, 2333619351, 9075135301, 35345312513, 137846529751, 538258059199, 2104098963721, 8233431436745, 32247603683171

Offset: 1

Vladeta Jovovic, Jan 17 2002

nonn

Cf. A034731, A052854.

Cf. A051731, A000984.

Table[Sum[Binomial[2*d-2,d-1], {d, Divisors[n]}], {n,1,30}] (* Vaclav Kotesovec, Jun 08 2019 *)

a(n)=if(n<1,0,sumdiv(n,d,binomial(2*d-2,d-1)))

a(n)=polcoeff(sum(m=1,n,x^m/sqrt(1-4*x^m+x*O(x^n))),n) /* Paul D. Hanna */

G.f.: Sum_{n>=1} x^n/sqrt(1-4*x^n). [From Paul D. Hanna, Aug 23 2011]
Logarithmic derivative of A052854, the number of unordered forests on n nodes.
Equals A051731 * A000984, i.e. the inverse Mobius transform of A000984. - Gary W. Adamson, Nov 09 2007
a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 08 2019

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A304787 Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).

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A052805 If B is a collection in which there are C(n-1) [Catalan numbers, A000108] things with n points, a(n) is the number of subsets without repetition of B with a total of n points.

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A246949 Decimal expansion of the coefficient K appearing in the asymptotic expression of the number of forests of ordered trees on n total nodes as K*4^(n-1)/sqrt(Pi*n^3).

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A066768 Sum_{d|n} binomial(2*d-2,d-1).

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PARI

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A246949 Decimal expansion of the coefficient K appearing in the asymptotic expression of the number of forests of ordered trees on n total nodes as K4^(n-1)/sqrt(Pin^3).