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A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Second bisection of A185251; the first bisection is A002699.

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

LINKS

Table of n, a(n) for n=0..24.

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)]

CROSSREFS

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

Sequence in context: A144646 A308417 A277520 * A000544 A221777 A227995

Adjacent sequences:  A303599 A303600 A303601 * A303603 A303604 A303605

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, May 09 2018

STATUS

approved

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Last modified August 5 06:18 EDT 2021. Contains 346457 sequences. (Running on oeis4.)