A271785 a(n) = Sum_{k=0..(n-1)/2} (n+2-k)*binomial(n-1-k,k).

0, 3, 4, 9, 16, 30, 54, 97, 172, 303, 530, 922, 1596, 2751, 4724, 8085, 13796, 23478, 39858, 67517, 114140, 192603, 324454, 545714, 916536, 1537275, 2575204, 4308897, 7201912, 12025038, 20058990, 33430297, 55667596, 92622471, 153992954, 255842890

Offset: 0

Author

R. J. Mathar, Apr 14 2016

Keywords

• nonn
• easy

Comments

This is a sibling to the expansions A001629(n+1) = Sum_{k=0..(n-1)/2} (n-k) *binomial(n-1-k,k) and A226432(n+3) = Sum_{k=0..(n-1)/2} (n+1-k) *binomial(n-1-k,k).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A001629.

Programs

Maple

A271785 := proc(n)
    add( (n+2-k)*binomial(n-1-k,k),k=0..(n-1)/2) ;
end proc:

Mathematica

LinearRecurrence[{2,1,-2,-1},{0,3,4,9},40] (* Harvey P. Dale, May 05 2020 *)

PARI

concat(0, Vec(x*(3-2*x-2*x^2)/(1-x-x^2)^2 + O(x^50))) \\ Colin Barker, Apr 14 2016

Formula

G.f.: x*(3-2*x-2*x^2) / (1-x-x^2)^2.

a(n) = 3*A001629(n+1) -2*A001629(n) -2*A001629(n-1).

From Colin Barker, Apr 14 2016: (Start)

a(n) = (2^(-1-n)*(-24*sqrt(5)*((1-sqrt(5))^n-(1+sqrt(5))^n)+5*((1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))*n))/25.

a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4) for n>3. (End)

E.g.f.: (1/25)*(sqrt(5)*(5*x + 24)*sinh((sqrt(5)*x)/2) + 15*x*cosh((sqrt(5)*x)/2))*exp(x/2). - Ilya Gutkovskiy, Apr 14 2016

a(n) = A006355(n+1)+A001629(n+1). - R. J. Mathar, May 20 2016