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 A137219 #9 Jan 05 2022 18:16:50
%S A137219 1,62,2649,116360,5364701,256452714,12582472897,629389744448,
%T A137219 31955247002601,1641724953315062,85159811841234857,
%U A137219 4452782349569991736,234393562418967430389,12409423916979629786322,660253088667210584565249
%N A137219 a(n) = (A126086(n) - 3*A001850(n) + 2)/6.
%H A137219 G. C. Greubel, <a href="/A137219/b137219.txt">Table of n, a(n) for n = 1..500</a>
%F A137219 a(n) = Sum_{m >= 0} binomial(binomial(m, n), 3)/2^(m+1).
%p A137219 A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137219 := proc(n) (A126086(n)-3*A001850(n)+2)/6 ; end: seq(A137219(n),n=1..30) ; # _R. J. Mathar_, Apr 01 2008
%t A137219 A126086[n_]:= A126086[n]= Sum[(-1)^k*Binomial[n+k,n]*HypergeometricPFQ[{-k, n+1, n+1}, {1,1}, 1], {k,0,2*n}];
%t A137219 A001850[n_]:= Hypergeometric2F1[-n, n+1, 1, -1];
%t A137219 A137219[n_]:= (A126086[n] - 3*A001850[n] + 2)/6;
%t A137219 Table[A137219[n], {n, 30}] (* _G. C. Greubel_, Jan 05 2022 *)
%o A137219 (Sage)
%o A137219 def A137219(n): return round( sum( binomial(binomial(j, n), 3)/2^(j+1) for j in (0..1000) ) )
%o A137219 [A137219(n) for n in (1..30)] # _G. C. Greubel_, Jan 05 2022
%Y A137219 Cf. A001850, A047665, A126086, A137220.
%K A137219 easy,nonn
%O A137219 1,2
%A A137219 _Vladeta Jovovic_, Mar 06 2008, Mar 16 2008
%E A137219 More terms from _R. J. Mathar_, Apr 01 2008