A103138 Second column of inverse of Delannoy triangle.
0, 1, -3, 10, -38, 158, -698, 3218, -15310, 74614, -370610, 1869338, -9549174, 49302030, -256859754, 1348695330, -7129819038, 37916710374, -202708895330, 1088819681834, -5873129780422, 31800514324606, -172780691083034, 941714095635890, -5147414826440558, 28210011946820438
Offset: 0
Examples
G.f.: A(x) = x - 3*x^2 + 10*x^3 - 38*x^4 + 158*x^5 - 698*x^6 + ... where A( x*(1+x)/(1-x) ) / (1-x) = x.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
CoefficientList[Series[(1-x*(1+x-(1+6*x+x^2)^(1/2))/(-2*x))*x*(1+x-(1+6*x+x^2)^(1/2))/(-2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *) -
PARI
{a(n)=if(n==1, 1, -polcoeff(sum(k=1, n-1, a(k)*x^k*(1+x)^k/(1-x+x*O(x^n))^(k+1)), n))} \\ Paul D. Hanna, Aug 06 2013
Formula
G.f.: (1-x*S(-x))*x*S(-x), where S(x) is the g.f. of the large Schroeder numbers A006318.
Conjecture: 2*n*a(n) +(13*n-20)*a(n-1) +(8*n-27)*a(n-2) +(n-5)*a(n-3)=0. - R. J. Mathar, Dec 14 2011
G.f.: x = Sum_{n>=1} a(n) * x^n * (1+x)^n / (1-x)^(n+1). - Paul D. Hanna, Aug 06 2013
G.f. satisfies: A(x*(1+x)/(1-x)) = x - x^2. - Paul D. Hanna, Aug 06 2013
a(n) ~ (-1)^n * (1-2*sqrt(2)) * sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
Comments