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A081698 Expansion of (1 - sqrt( 1 - 4*x*sqrt( 1 + 4*x )) )/( 2*x ). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

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

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-Fran├žois 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 *)

CROSSREFS

Cf. A081696.

Sequence in context: A034475 A156173 A094632 * A182096 A012123 A012255

Adjacent sequences:  A081695 A081696 A081697 * A081699 A081700 A081701

KEYWORD

easy,nonn

AUTHOR

Emanuele Munarini, Apr 02 2003

STATUS

approved

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Last modified March 28 20:44 EDT 2020. Contains 333103 sequences. (Running on oeis4.)