cp's OEIS Frontend

A277924 a(n) = Sum_{i=0..n+1} binomial(2*n,n-i+1).

%I A277924 #33 Dec 28 2023 07:09:12
%S A277924 1,4,15,57,219,848,3302,12911,50643,199140,784626,3096514,12236830,
%T A277924 48412432,191718188,759852347,3013746563,11960699132,47494802618,
%U A277924 188689585982,749961486698,2981943800192,11860758904148,47191458566582
%N A277924 a(n) = Sum_{i=0..n+1} binomial(2*n,n-i+1).
%C A277924 a(n) = A124234(2*n+1,n) is diagonal of triangle A124234.
%H A277924 Seiichi Manyama, <a href="/A277924/b277924.txt">Table of n, a(n) for n = 0..1661</a>
%F A277924 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)).
%F A277924 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
%F A277924 a(n) = ((3*n+1)*Catalan(n)+4^n)/2. - _Peter Luschny_, Nov 04 2016
%p A277924 a:= proc(n) option remember; `if`(n<3,
%p A277924       [1, 4, 15][n+1], ((2*(4*n^2-7*n-1))*a(n-1)
%p A277924       -(8*(n-1))*(2*n-3)*a(n-2))/((n-2)*(n+1)))
%p A277924     end:
%p A277924 seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 04 2016
%t A277924 Table[((3n + 1) CatalanNumber[n] + 4^n)/2, {n,0,23}] (* _Peter Luschny_, Nov 04 2016 *)
%o A277924 (Maxima) makelist(sum(binomial(2*n, n-i+1), i, 0, n+1), n, 0, 30);
%o A277924 (PARI) a(n) = sum(k=0, n+1, binomial(2*n, n-k+1)); \\ _Michel Marcus_, Nov 04 2016
%o A277924 (Sage)
%o A277924 def a():
%o A277924     c, f, b, n = 1, 1, 1, 1
%o A277924     while True:
%o A277924         yield (c*f + b)//2
%o A277924         c = (4*n - 2)*c//(n + 1)
%o A277924         b <<= 2
%o A277924         f += 3
%o A277924         n += 1
%o A277924 A277924 = a()
%o A277924 print([next(A277924) for _ in range(23)]) # _Peter Luschny_, Nov 04 2016
%Y A277924 Cf. A000108, A124234.
%K A277924 nonn
%O A277924 0,2
%A A277924 _Vladimir Kruchinin_, Nov 04 2016