A185087 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A000108(k+1).

1, 1, 3, 5, 12, 24, 55, 119, 272, 612, 1411, 3247, 7565, 17667, 41561, 98099, 232696, 553784, 1322813, 3169065, 7614583, 18342921, 44294991, 107200829, 259983346, 631718606, 1537737567, 3749440151, 9156561590, 22394270034, 54845701243

Offset: 0

Author

Paul Barry, Feb 18 2011

Keywords

• nonn
• easy

Comments

Essentially identical to A090345 (=1,0,1,1,3,5,12,24...). - Joerg Arndt, Mar 18 2011

Hankel transform is A008619(n+1) (counting numbers doubled).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Mathematica

CoefficientList[Series[(1 - x - 2*x^2 - Sqrt[1 - 2 x - 3 x^2 + 4 x^3])/(2*x^4), {x,0,50}], x] (* G. C. Greubel, Jun 22 2017 *)

PARI

x='x+O('x^50); Vec((1-x-2*x^2-sqrt(1-2*x-3*x^2+4*x^3))/(2*x^4)) \\ G. C. Greubel, Jun 22 2017

Formula

G.f.: (1-x-2x^2-sqrt(1-2x-3x^2+4x^3))/(2x^4).

G.f.: 1/(1-x-2x^2/(1-(1/2)x^2/(1-x-(3/2)x^2/(1-(2/3)x^2/(1-x-(4/3)x^2/(1-(3/4)x^2/(1-... (continued fraction).

a(n)=sum{k=0..n, sum{j=0..n, binomial(k-j,n-k-j)*binomial(k,j)*if(n-k-j>=0, A001006(n-k-j),0)}}.

a(n)=A090345(n+2).

Conjecture: (n+4)*a(n) -(2*n+5)*a(n-1) -3*(n+1)*a(n-2) +2*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 16 2011