A029887 A sum over scaled A000531 related to Catalan numbers C(n).
1, 11, 82, 515, 2934, 15694, 80324, 397923, 1922510, 9105690, 42438076, 195165646, 887516252, 3997537980, 17857602568, 79200753059, 349051186494, 1529735010658, 6670733733260, 28959032959962, 125209652884756, 539384745200516, 2315840230811832, 9912689725127950
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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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
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Mathematica
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 *) CoefficientList[Series[(4 x - 1 + Sqrt[1 - 4 x])/(2 x (1 - 4 x)^3), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 14 2014 *) -
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
Formula
a(n) = 4^n * Sum_{k=0..n} A000531(k+1)/4^k.
a(n) = (1/3)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n) - (n+2)*2^(2*n+1).
a(n) = 4*a(n-1) + A000531(n+1).
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.
G.f.: (4*x-1+sqrt(1-4*x))/(2*x*(1-4*x)^3). - Vincenzo Librandi, Mar 14 2014
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (1/24)*(n+2)*((n+3)*(n+4)*Catalan(n+3) - 3*4^(n+2)).
a(n) = (1/2)*A000184(n+2). (End)
Extensions
More terms from Vincenzo Librandi, Mar 14 2014
Comments