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A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

%I A137220 #20 Jan 06 2022 02:30:53
%S A137220 1,4,75,2712,116681,5366384,256461703,12582521536,629390010177,
%T A137220 31955248465164,1641724961412515,85159811886281576,
%U A137220 4452782349821587705,234393562420377364008,12409423916987553634575,660253088667255226947072
%N A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.
%H A137220 Andrew Howroyd, <a href="/A137220/b137220.txt">Table of n, a(n) for n = 0..200</a>
%F A137220 a(n) = -Sum_{m>=0} binomial(-binomial(m,n),3)/2^(m+1).
%F A137220 a(n) = A137219(n) + A001850(n). - _R. J. Mathar_, Apr 01 2008
%F A137220 a(n) = Sum_{j=0..3*n} binomial(binomial(j,n)+2, 3) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - _Andrew Howroyd_, Feb 09 2020
%p A137220 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: A137220 := proc(n) (A126086(n)+3*A001850(n)+2)/6 ; end: seq(A137220(n),n=0..30) ; # _R. J. Mathar_, Apr 01 2008
%t A137220 T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
%t A137220 Table[T[n, 3], {n, 0, 15}] (* _Jean-François Alcover_, Apr 10 2020, after _Andrew Howroyd_ *)
%o A137220 (PARI) a(n) = {sum(j=0, 3*n, binomial(binomial(j,n)+2, 3) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ _Andrew Howroyd_, Feb 09 2020
%o A137220 (Sage)
%o A137220 @CachedFunction
%o A137220 def A137220(n): return round( -sum( binomial(-binomial(j, n), 3)/2^(j+1) for j in (0..500) ) )
%o A137220 [A137220(n) for n in (0..30)] # _G. C. Greubel_, Jan 05 2022
%Y A137220 Column k=3 of A330942.
%Y A137220 Cf. A047665, A137219.
%K A137220 easy,nonn
%O A137220 0,2
%A A137220 _Vladeta Jovovic_, Mar 06 2008, Mar 16 2008
%E A137220 More terms from _R. J. Mathar_, Apr 01 2008