A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108).

1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, 1105478, 6227712, 35520498, 204773400, 1191572004, 6990859416, 41313818217, 245735825082, 1470125583756, 8840948601024, 53417237877396, 324123222435804, 1974317194619712

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

Conjecture D-finite with recurrence 1053*n*(n-1)*(n-2)*(n-3)*a(n) -36*(n-1)*(n-2)*(n-3)*(634*n-1367)*a(n-1) +24*(n-2)*(n-3)*(7966*n^2-43500*n+61181)*a(n-2) -8*(n-3)*(96128*n^3-957424*n^2+3221878*n-3665189)*a(n-3) +16*(91904*n^4-1446528*n^3+8575792*n^2-22703688*n+22652013)*a(n-4) -256*(8*n-35)*(8*n-41)*(8*n-39)*(8*n-37)*a(n-5)=0. - R. J. Mathar, Aug 30 2021

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 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 range(k, n + 1)) for k in range(1, n + 1)) / n
[a(n) for n in range(1, 51)] # Indranil Ghosh, Apr 12 2017

Crossrefs

Cf. A121988 (2nd), A158825, A158827 (4th), A158828, A158829.

Sequence in context: A125188 A054666 A006026 * A107264 A370441 A200740

Adjacent sequences: A158823 A158824 A158825 * A158827 A158828 A158829

Keyword

nonn

Author

Paul D. Hanna, Mar 28 2009

Status

approved