A081698 Expansion of (1 - sqrt( 1 - 4*x*sqrt( 1 + 4*x )) )/( 2*x ).

1, 3, 4, 21, 56, 282, 984, 4813, 19280, 93150, 403672, 1945954, 8845360, 42766292, 200419504, 974134461, 4659558048, 22785183670, 110564976792, 543935554390, 2667398588272, 13196971915628, 65238895435792, 324431740601618, 1614044041864800, 8063536826420460

Offset: 0

Author

Emanuele Munarini, Apr 02 2003

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A081696.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 3, 4, 21][n+1],
      (2*n*(n+1)*(3-2*n) *a(n-1) +4*n*(2*n-1)*(2*n-3) *a(n-2)
       +8*(2*n-3)*(8*n^2-16*n-15) *a(n-3)
       +16*(4*n-15)*(4*n-9)*(n+1) *a(n-4)) /(n^2*(n+1)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 13 2013

Mathematica

a[n_] := Sum[Binomial[(k+1)/2, n-k]*Binomial[2*k, k]*4^(n-k)/(k+1), {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2015, after Vladimir Kruchinin *)
CoefficientList[Series[(1-Sqrt[1-4x Sqrt[1+4x]])/(2x),{x,0,30}],x] (* Harvey P. Dale, Oct 30 2017 *)

Formula

G.f.: (1-sqrt(1-4*x*sqrt(1+4*x)))/(2*x).

a(n) = sum(k=0..n, (binomial((k+1)/2,n-k)*binomial(2*k,k)*4^(n-k))/(k+1)). [Vladimir Kruchinin, Mar 13 2013]

D-finite with recurrence: n*(n+1)*a(n) +2*n*(5*n-7)*a(n-1) +4*(2*n^2-13*n+12)*a(n-2) -8*(2*n-3)*(14*n-37)*a(n-3) +16*(-64*n^2+392*n-573)*a(n-4) -96*(4*n-13)*(4*n-19)*a(n-5)=0. - R. J. Mathar, Jan 23 2020