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 A202948 #10 Feb 16 2025 08:33:16
%S A202948 -3,-72,-7560,-1814400,-778377600,-523069747200,-506854585036800,
%T A202948 -669048052248576000,-1154107890128793600000,
%U A202948 -2520571632041285222400000,-6797981691615346244812800000
%N A202948 a(n+1) = 3*A136016*a(n).
%C A202948 Sums of coefficients from (3n+2)th moments of binomial(m,k)*binomial(2m,k): see Maple code below.
%H A202948 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/BinomialSums.html">MathWorld: Binomial Sums</a>
%F A202948 a(n)=-(1/6)*27^n*GAMMA(n-1/3)*GAMMA(n+1/3)*sqrt(3)/Pi.
%e A202948 The evaluation of sum(k^8 binomial(n,k) binomial(2n,k), k=0..n) involves the polynomial 64n^10 + 192n^9 - 1344n^8 - 1056n^7 + 8256n^6 - 3696n^5 - 9940n^4 + 7551n^3 + 348n^2 - 507n + 60, the sum of the coefficients of which is -72=a(2).
%p A202948 with(PolynomialTools); polyn := proc (q) options operator, arrow; 3^q*Pi*GAMMA(2*n)*(sum(k^q*binomial(n, k)*binomial(2*n, k), k = 0 .. n))/(27^n*sqrt(3)*GAMMA(n-floor((1/3)*q+1/3)+2/3)*GAMMA(n-floor((1/3)*q)+1/3)) end proc; coefl := proc (q) options operator, arrow; CoefficientList(expand(polyn(q)), n) end proc; coe := proc (j, h) options operator, arrow; coefl(j)[h] end proc; seq(sum(coe(3*r+2, k), k = 1 .. 5*r+3), r = 1 .. 8) ;
%o A202948 (PARI) print1(a=-3);for(n=2,20,print1(", ",a*=27*n*(n-2)+24)) \\ _Charles R Greathouse IV_, Dec 27 2011
%Y A202948 Cf. A136016, A064350.
%K A202948 sign,easy
%O A202948 1,1
%A A202948 _John M. Campbell_, Dec 26 2011