A277924 a(n) = Sum_{i=0..n+1} binomial(2*n,n-i+1).
1, 4, 15, 57, 219, 848, 3302, 12911, 50643, 199140, 784626, 3096514, 12236830, 48412432, 191718188, 759852347, 3013746563, 11960699132, 47494802618, 188689585982, 749961486698, 2981943800192, 11860758904148, 47191458566582
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1661
Programs
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Maple
a:= proc(n) option remember; `if`(n<3, [1, 4, 15][n+1], ((2*(4*n^2-7*n-1))*a(n-1) -(8*(n-1))*(2*n-3)*a(n-2))/((n-2)*(n+1))) end: seq(a(n), n=0..25); # Alois P. Heinz, Nov 04 2016 -
Mathematica
Table[((3n + 1) CatalanNumber[n] + 4^n)/2, {n,0,23}] (* Peter Luschny, Nov 04 2016 *) -
Maxima
makelist(sum(binomial(2*n, n-i+1), i, 0, n+1), n, 0, 30);
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PARI
a(n) = sum(k=0, n+1, binomial(2*n, n-k+1)); \\ Michel Marcus, Nov 04 2016
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Sage
def a(): c, f, b, n = 1, 1, 1, 1 while True: yield (c*f + b)//2 c = (4*n - 2)*c//(n + 1) b <<= 2 f += 3 n += 1 A277924 = a() print([next(A277924) for in range(23)]) # _Peter Luschny, Nov 04 2016
Formula
G.f.: x^2*(1 - 2*x - sqrt(1 - 4*x))/(sqrt(1 - 4*x)*(4*x^3 - 13*x^2 + 7*x - 1) - 20*x^3 + 25*x^2 - 9*x + 1) - 1/x = (-1 + 5*x + (1 - x)*sqrt(1 - 4*x))/(2*x*(1 - 4*x)).
a(n) = 2^(2*n-1) - 4^n*(n-1/2)!/(sqrt(Pi)*(n+1)!) + 3*2^(2*n-1)*(n-1/2)!/(sqrt(Pi)*n!). - Bruno Berselli, Nov 04 2016
a(n) = ((3*n+1)*Catalan(n)+4^n)/2. - Peter Luschny, Nov 04 2016
Comments