cp's OEIS Frontend

A029887 A sum over scaled A000531 related to Catalan numbers C(n).

%I A029887 #24 Aug 15 2025 21:06:47
%S A029887 1,11,82,515,2934,15694,80324,397923,1922510,9105690,42438076,
%T A029887 195165646,887516252,3997537980,17857602568,79200753059,349051186494,
%U A029887 1529735010658,6670733733260,28959032959962,125209652884756,539384745200516,2315840230811832,9912689725127950
%N A029887 A sum over scaled A000531 related to Catalan numbers C(n).
%C A029887 Related to planar maps? - see A000184. - _N. J. A. Sloane_, Mar 11 2007
%H A029887 Vincenzo Librandi, <a href="/A029887/b029887.txt">Table of n, a(n) for n = 0..1000</a>
%F A029887 a(n) = 4^n * Sum_{k=0..n} A000531(k+1)/4^k.
%F A029887 a(n) = (1/3)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n) - (n+2)*2^(2*n+1).
%F A029887 a(n) = 4*a(n-1) + A000531(n+1).
%F A029887 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 A029887 G.f.: (4*x-1+sqrt(1-4*x))/(2*x*(1-4*x)^3). - _Vincenzo Librandi_, Mar 14 2014
%F A029887 From _G. C. Greubel_, Jul 18 2024: (Start)
%F A029887 a(n) = (1/24)*(n+2)*((n+3)*(n+4)*Catalan(n+3) - 3*4^(n+2)).
%F A029887 a(n) = (1/2)*A000184(n+2). (End)
%t A029887 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 A029887 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 A029887 (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 A029887 (SageMath) [(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 A029887 Cf. A000108, A000184, A000531, A002802.
%K A029887 nonn,easy
%O A029887 0,2
%A A029887 _Wolfdieter Lang_
%E A029887 More terms from _Vincenzo Librandi_, Mar 14 2014