A104532 Expansion of (1+sqrt(1-4*x))/(6*sqrt(1-4*x)-4).

1, 5, 35, 250, 1795, 12910, 92910, 668820, 4815075, 34667110, 249598330, 1797091180, 12938997710, 93160575500, 670755400700, 4829436210600, 34771931021475, 250357867996950, 1802576519933250, 12978550465880700, 93445561587077850, 672808036862840100, 4844217840946181700

Offset: 0

Author

Paul Barry, Mar 12 2005

Keywords

• easy
• nonn

Comments

Apply the Riordan matrix ((1+sqrt(1-4x))/2,(1-sqrt(1-4x))/2) (inverse (1/(1-x),x(1-x))) to 6^n. In general, (1+sqrt(1-4x))(k*sqrt(1-4x)-(k-2)) results from applying the Riordan matrix ((1+sqrt(1-4x))/2,(1-sqrt(1-4x))/2) (inverse of (1/(1-x),x(1-x))) to k^n. We then have a(n)=0^n+sum{i=0..n, (k-1)^(i+1)*C(2n-1,n-i-1)2(i+1)/(n+i+1)}.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A088218, A067336, A076025, 104530, 104531.

Programs

Mathematica

CoefficientList[Series[(1+Sqrt[1-4x])/(6Sqrt[1-4x]-4), {x,0,20}], x]  (* Harvey P. Dale, Apr 11 2011 *)

PARI

x='x+O('x^66); Vec((1+sqrt(1-4*x))/(6*sqrt(1-4*x)-4)) \\ Joerg Arndt, May 13 2013

Formula

a(n) = 0^n + sum{k=0..n, 5^(k+1)*C(2n-1, n-k-1)*2*(k+1)/(n+k+1)}.

D-finite with recurrence: 5*n*a(n) = 2*(28*n-15)*a(n-1) - 72*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 2^(2*n+1)*3^(2*n-1)/5^n. - Vaclav Kotesovec, Oct 17 2012