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
Programs
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Maple
seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # Paolo P. Lava, May 10 2018
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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
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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).
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