This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A303602 #40 Oct 23 2023 11:55:23
%S A303602 0,3,25,154,837,4246,20618,97140,447661,2028478,9070110,40122028,
%T A303602 175913250,765561564,3310623412,14238676712,60949133949,259809601870,
%U A303602 1103420316566,4670886541308,19714134528598,82985455688276,348481959315660,1460179866076504,6106070639175122
%N A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).
%C A303602 Second bisection of A185251; the first bisection is A002699.
%C A303602 The terms are not congruent to 5 (mod 6).
%F A303602 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.
%F A303602 From _Vaclav Kotesovec_, May 10 2018: (Start)
%F A303602 G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
%F A303602 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)
%F A303602 a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
%F A303602 a(n+1) - 4*a(n) = A164991(2*n+3).
%p A303602 seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # _Paolo P. Lava_, May 10 2018
%t A303602 Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
%t A303602 CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* _Vaclav Kotesovec_, May 10 2018 *)
%o A303602 (Sage) [(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]
%o A303602 (PARI) a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A303602 Cf. A000346, A002699, A005408, A047226, A164991, A185251.
%K A303602 nonn,easy
%O A303602 0,2
%A A303602 _Bruno Berselli_, May 09 2018