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Sum(C(j)*(n-j)*4^(n-j),j=0..n-1), C = Catalan numbers.
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%I #20 Jul 17 2023 15:37:14

%S 0,4,36,232,1300,6744,33320,159184,742068,3395320,15308920,68213424,

%T 300999816,1317415792,5726300880,24742452128,106357582324,

%U 455122855224,1939780103768,8238185701360,34876073003352,147223869286736,619871651308336,2603757232133472,10913483674589000

%N Sum(C(j)*(n-j)*4^(n-j),j=0..n-1), C = Catalan numbers.

%H Vincenzo Librandi, <a href="/A018217/b018217.txt">Table of n, a(n) for n = 0..1000</a>

%F Recurrence: (n-1)*n*a(n) = 2*(n-1)*(4*n+1)*a(n-1) - 8*n*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 07 2012

%F G.f.: 2/(1-4*x)^2 - 2/(1-4*x)^(3/2). - _Mark van Hoeij_, Mar 28 2013

%t Table[2*(n+1)*(4^n-Binomial[2*n+1,n]),{n,0,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)

%t Table[Sum[CatalanNumber[j](n-j)4^(n-j),{j,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Jul 17 2023 *)

%o (PARI) x='x+O('x^66); concat([0], Vec( 2/(1-4*x)^2 - 2/(1-4*x)^(3/2) ) ) \\ _Joerg Arndt_, May 04 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Peter Winkler (pw(AT)bell-labs.com)