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%I #26 Jan 30 2020 21:29:14
%S 1,4,11,27,66,170,471,1398,4381,14282,47897,164012,570639,2010678,
%T 7158569,25709157,93020112,338736224,1240496193,4565563209,
%U 16878057692,62644246662,233346693759,872045012633,3268643350608,12285088109136,46288732360369,174813127020311,661606223839322
%N Apply partial sum operator thrice to Catalan numbers.
%H Vincenzo Librandi, <a href="/A014151/b014151.txt">Table of n, a(n) for n = 0..1000</a>
%F D-finite with recurrence: n*(n+1)*a(n) = 2*n*(3*n+1)*a(n-1) - (9*n^2+7*n-4)*a(n-2) + 2*(n+1)*(2*n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) ~ 2^(2*n+6)/(27*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F G.f.: C(x)/(1-x)^3, where C(x) is g.f. of Catalan numbers. - _Vladimir Kruchinin_, Oct 18 2016
%F a(n) = Sum_{k=0..n} binomial(n+3,k+3) * r(k), r(k) = A005043(k). - _Vladimir Kruchinin_, Oct 18 2016
%t Flatten[{1, RecurrenceTable[{n*(n+1)*a[n] == 2*n*(3*n+1)*a[n-1] - (9*n^2+7*n-4)*a[n-2] + 2*(n+1)*(2*n+1)*a[n-3],a[1]==4,a[2]==11,a[3]==27},a,{n,100}]}]
%o (PARI)
%o sm(v)={my(s=vector(#v));s[1]=v[1];for(n=2,#v,s[n]=v[n]+s[n-1]);s;}
%o C(n)=binomial(2*n,n)/(n+1);
%o sm(sm(sm(vector(66,n,C(n-1)))))
%o /* _Joerg Arndt_, May 04 2013 */
%Y Cf. A000108, A014137, A005043
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.
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