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A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108). 5
1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, 1105478, 6227712, 35520498, 204773400, 1191572004, 6990859416, 41313818217, 245735825082, 1470125583756, 8840948601024, 53417237877396, 324123222435804, 1974317194619712 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Series reversion of x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8. - Benedict W. J. Irwin, Oct 19 2016

Column 1 of A106566^3 (see Barry, Section 3). - Peter Bala, Apr 11 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

FORMULA

a(n) = (1/n)*Sum_{k=1..n} [ binomial(2*k-2,k-1)*Sum_{i=k..n}( binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1) ) ]. - Vladimir Kruchinin, Jan 24 2013

G.f.: (1 - sqrt(-1 + 2*sqrt(-1 + 2*sqrt(1 - 4*x))))/2. - Benedict W. J. Irwin, Oct 19 2016

a(n) ~ 2^(8*n - 3) / (sqrt(5*Pi) * n^(3/2) * 39^(n - 1/2)). - Vaclav Kotesovec, Jul 20 2019

MATHEMATICA

max = 22; c[x_] := Sum[ CatalanNumber[n]*x^n, {n, 0, max}]; f[x_] := x*c[x]; CoefficientList[ Series[ f@f@f@x, {x, 0, max}], x] // Rest (* Jean-François Alcover, Jan 24 2013 *)

Rest@CoefficientList[InverseSeries[x-3x^2+6x^3-9x^4+10x^5-8x^6+4x^7-x^8+O[x]^30], x] (* Benedict W. J. Irwin, Oct 19 2016 *)

PROG

(PARI) a(n)=local(F=serreverse(x-x^2+O(x^(n+1))), G=x); for(i=1, 3, G=subst(F, x, G)); polcoeff(G, n)

(Maxima)

a(n):=sum(binomial(2*k-2, k-1)*sum(binomial(-k+2*i-1, i-1)*binomial(2*n-i-1, n-1), i, k, n), k, 1, n)/n; // Vladimir Kruchinin, Jan 24 2013

(Python)

from sympy import binomial as C

def a(n): return (1/n)*sum([C(2*k - 2, k - 1) * sum([C(-k + 2*i - 1, i - 1) * C(2*n - i - 1, n - 1) for i in xrange(k, n + 1)]) for k in xrange(1, n + 1)])

print [a(n) for n in xrange(1, 51)] # Indranil Ghosh, Apr 12 2017

CROSSREFS

Cf. A158825, A158827, A158828, A158829.

Sequence in context: A125188 A054666 A006026 * A107264 A200740 A177133

Adjacent sequences:  A158823 A158824 A158825 * A158827 A158828 A158829

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 28 2009

STATUS

approved

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Last modified November 19 18:39 EST 2019. Contains 329323 sequences. (Running on oeis4.)