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A217615 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

%I A217615 #36 Feb 19 2025 05:44:05
%S A217615 1,1,1,3,5,7,15,29,49,95,187,345,659,1289,2465,4739,9237,17911,34715,
%T A217615 67705,132063,257477,503309,984983,1927895,3778017,7411237,14544967,
%U A217615 28565661,56144615,110406527,217225533,427636561,842256047,1659600955,3271579689,6451913519
%N A217615 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).
%C A217615 Radius of convergence of g.f. is r = 1/2.
%C A217615 More generally, given
%C A217615 A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
%C A217615 then A(x) = 1/sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
%C A217615 and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t > 0.
%C A217615 a(n) is the number of (2k-1)-element subsets of {1, 2, ..., n+1} whose k-th smallest (i.e., k-th largest) element equals 2k-1. - _Darij Grinberg_, Oct 09 2019
%H A217615 G. C. Greubel, <a href="/A217615/b217615.txt">Table of n, a(n) for n = 0..1000</a>
%F A217615 G.f.: A(x) = 1 / sqrt( (1 - x + 2*x^2)^2 - 4*x^2 ).
%F A217615 G.f.: A(x) = 1 / sqrt( (1-x)*(1-2*x)*(1+x+2*x^2) ).
%F A217615 G.f. satisfies: A(x) = (1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)) / (1-x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
%F A217615 a(n) ~ 2^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Sep 16 2013
%F A217615 a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k) * binomial(n-2*k, k). - _Darij Grinberg_, Oct 09 2019
%F A217615 a(n) = hypergeom([1/2,(1-n)/3,(2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27) for n >= 4. - _Peter Luschny_, Oct 09 2019
%e A217615 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 15*x^6 + 29*x^7 + 49*x^8 + ...
%e A217615 where the g.f. equals the series:
%e A217615 A(x) = 1 +
%e A217615   x*((1-x) + x) +
%e A217615   x^2*((1-x)^2 + 2^2*x*(1-x) + x^2) +
%e A217615   x^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3) +
%e A217615   x^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4) +
%e A217615   x^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) + ...
%p A217615 a := n -> `if`(n < 4, [1, 1, 1, 3][n+1], hypergeom([1/2, (1-n)/3, (2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27)):
%p A217615 seq(simplify(a(n)), n=0..36); # _Peter Luschny_, Oct 09 2019
%t A217615 CoefficientList[Series[1/Sqrt[(1-x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 16 2013 *)
%o A217615 (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
%o A217615 for(n=0,40,print1(a(n),", "))
%o A217615 (PARI) a(n)={sum(k=0, n\2, binomial(2*k, k) * binomial(n-2*k, k))} \\ _Andrew Howroyd_, Oct 09 2019
%Y A217615 Cf. A217616, A217617, A217461, A217464, A216604, A217661.
%K A217615 nonn
%O A217615 0,4
%A A217615 _Paul D. Hanna_, Oct 09 2012