A071357 Expansion of (1 - 4*x - (1-2*x)*sqrt(1-4*x-4*x^2))/(8*x^3).

0, 1, 4, 16, 64, 260, 1072, 4480, 18944, 80928, 348800, 1515008, 6625280, 29147456, 128918272, 572928000, 2557100032, 11457170944, 51514963968, 232370167808, 1051235287040, 4768568354816, 21684663148544, 98835356778496, 451433970008064

Offset: 0

Author

N. J. A. Sloane, Jun 12 2002

Keywords

• nonn

Links

• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Crossrefs

Cf. A000108, A065096.

Programs

Mathematica

CoefficientList[Series[(1-4x-(1-2x)Sqrt[1-4x-4x^2])/(8x^3),{x,0,30}],x] (* Harvey P. Dale, Aug 09 2016 *)
Table[Sum[2^(n-k) * Binomial[n, k] * 2^((k-1)/2) * CatalanNumber[(k+1)/2] * (1 - (-1)^k)/2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 03 2019 *)

Formula

Binomial transform is A065096. - Paul Barry, Sep 16 2006

a(n) = (1/Pi)*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*sqrt(-x^2+4x+4)*(x-2)/8. - Paul Barry, Sep 16 2006

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(n,k)*2^((k-1)/2)*C((k-1)/2+1)*(1-(-1)^k)/2, where C(n)=A000108(n). - Paul Barry, Sep 16 2006

D-finite with recurrence: a(n) = (1/(n+3))*((6*n+8)*a(n-1) - (4*n-4)*a(n-2) - (8*n-16)*a(n-3)) for n > 2, with a(0)=0, a(1)=1, a(2)=4. - Tani Akinari, Jul 04 2013

a(n) ~ 2^(n + 1/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019