A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).

0, 3, 25, 154, 837, 4246, 20618, 97140, 447661, 2028478, 9070110, 40122028, 175913250, 765561564, 3310623412, 14238676712, 60949133949, 259809601870, 1103420316566, 4670886541308, 19714134528598, 82985455688276, 348481959315660, 1460179866076504, 6106070639175122

Offset: 0

Author

Bruno Berselli, May 09 2018

Keywords

• nonn
• easy

Comments

Second bisection of A185251; the first bisection is A002699.

The terms are not congruent to 5 (mod 6).

Crossrefs

Cf. A000346, A002699, A005408, A047226, A164991, A185251.

Programs

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 *)

PARI

a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ Charles R Greathouse IV, Oct 23 2023

Sage

[(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]

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).