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 A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k). 2
 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)] (PARI) a(n)=(2*n+1)*(4^n-binomial(2*n, n))/2 \\ Charles R Greathouse IV, Oct 23 2023 CROSSREFS Cf. A000346, A002699, A005408, A047226, A164991, A185251. Sequence in context: A308417 A277520 A367507 * A000544 A356200 A221777 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 4 22:06 EDT 2024. Contains 374934 sequences. (Running on oeis4.)