A029887 - OEIS

%I #20 Jul 18 2024 04:49:30

%S 1,11,82,515,2934,15694,80324,397923,1922510,9105690,42438076,

%T 195165646,887516252,3997537980,17857602568,79200753059,349051186494,

%U 1529735010658,6670733733260,28959032959962,125209652884756,539384745200516,2315840230811832,9912689725127950

%N A sum over scaled A000531 related to Catalan numbers C(n).

%C Related to planar maps? - see A000184. - _N. J. A. Sloane_, Mar 11 2007

%H Vincenzo Librandi, <a href="/A029887/b029887.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 4^n * Sum_{k=0..n} A000531(k+1)/4^k.

%F a(n) = (1/3)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n) - (n+2)*2^(2*n+1).

%F a(n) = 4*a(n-1) + A000531(n+1).

%F G.f. c(x)/(1-4*x)^(5/2) = (2-c(x))/(1-4*x)^3, where c(x) = g.f. for Catalan numbers; also convolution of Catalan numbers with A002802.

%F G.f.: (4*x-1+sqrt(1-4*x))/(2*x*(1-4*x)^3). -  _Vincenzo Librandi_, Mar 14 2014

%F From _G. C. Greubel_, Jul 18 2024:

%F a(n) = (1/24)*(n+2)*((n+3)*(n+4)*Catalan(n+3) - 3*4^(n+2)).

%F a(n) = (1/2)*A000184(n+2). (End)

%t a[n_] := (2*n+1)*(2*n+3)*(2*n+5)*CatalanNumber[n]/3 - (n+2)*2^(2*n+1); Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Mar 12 2014 *)

%t CoefficientList[Series[(4 x - 1 + Sqrt[1 - 4 x])/(2 x (1 - 4 x)^3), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 14 2014 *)

%o (Magma) [(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n)/3 - (n+2)*2^(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, Mar 14 2014

%o (SageMath)

%o [(n+2)*((n+3)*(n+4)*catalan_number(n+3) - 3*4^(n+2))//24 for n in range(31)] # _G. C. Greubel_, Jul 18 2024

%Y Cf. A000108, A000184, A000531, A002802.

%K nonn

%O 0,2

%A _Wolfdieter Lang_

%E More terms from _Vincenzo Librandi_, Mar 14 2014