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
Keywords
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.
Programs
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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 *) -
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 */
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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)
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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
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
Comments