OFFSET
0,2
COMMENTS
FORMULA
E.g.f.: ((1 + 8*x)*exp(2*x) - (1 + 4*x)*I_0(2*x) - 4*x*I_1(2*x))*exp(2*x)/2, where I_m(.) is the modified Bessel function of the first kind.
From Vaclav Kotesovec, May 10 2018: (Start)
G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
D-finite with recurrence: n*(2*n-1)*a(n) = 2*(2*n+1)*(4*n-3)*a(n-1) - 8*(2*n-1)*(2*n+1)*a(n-2). (End)
a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
a(n+1) - 4*a(n) = A164991(2*n+3).
MAPLE
seq(add(k*binomial(2*n+1, k), k=0..n), n=0..24); # Paolo P. Lava, May 10 2018
MATHEMATICA
Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* Vaclav Kotesovec, May 10 2018 *)
PROG
(Sage) [(2*n+1)*(4^n-binomial(2*n, n))/2 for n in (0..30)]
(PARI) a(n)=(2*n+1)*(4^n-binomial(2*n, n))/2 \\ Charles R Greathouse IV, Oct 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, May 09 2018
STATUS
approved