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%I A152548 #28 Apr 02 2024 11:44:23
%S A152548 1,4,10,24,54,120,260,560,1190,2520,5292,11088,23100,48048,99528,
%T A152548 205920,424710,875160,1798940,3695120,7574996,15519504,31744440,
%U A152548 64899744,132503644,270415600,551231800,1123264800,2286646200,4653525600
%N A152548 Sum of squared terms in rows of triangle A152547: a(n) = Sum_{k=0..C(n,[n/2])-1} A152547(n,k)^2.
%F A152548 G.f.: A(x) = sqrt( (1+2x)/(1-2x)^3 ).
%F A152548 a(n) = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).
%F A152548 a(n) = A107233(n)/(n+1).
%F A152548 Self-convolution equals A014477.
%F A152548 E.g.f.: ((1 + 4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - _Peter Luschny_, Aug 26 2012
%F A152548 a(n) = (-2)^n*hypergeom([-n,3/2], [1], 2). - _Peter Luschny_, Apr 26 2016
%F A152548 D-finite with recurrence: (n+1)*a(n+1) = 4*a(n) + 4*n*a(n-1). - _Vladimir Reshetnikov_, Oct 10 2016
%F A152548 a(n) ~ 2^(n + 3/2) * sqrt(n/Pi). - _Vaclav Kotesovec_, Oct 11 2016
%F A152548 From _Peter Bala_, Mar 31 2024: (Start)
%F A152548 a(n) = (2^n) * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(-3/2, n-k).
%F A152548 a(n) = (2^n) * Sum_{k = 0..n} (2^k)*binomial(n, k)*binomial(1/2, k).
%F A152548 a(n) = (2^n)* Sum_{k = 0..n} binomial(n, k)*binomial(k+1/2, n). See A008288.
%F A152548 a(n) = (2*n + 1)!/(2^n * n!^2) * hypergeom([-n, -1/2], [-n-1/2], -1).
%F A152548 a(n) = 2^n * hypergeom([-n, -1/2], [1], 2).
%F A152548 a(n) = (-1/2)^n * binomial(2*n, n)/(1 - 2*n) * hypergeom([-n, 3/2], [-n+3/2], -1).(End)
%p A152548 seq(simplify((-2)^n*hypergeom([-n,3/2], [1], 2)),n=0..29); # _Peter Luschny_, Apr 26 2016
%t A152548 CoefficientList[Series[Sqrt[(1+2x)/(1-2x)^3],{x,0,30}],x] (* _Harvey P. Dale_, Jan 04 2016 *)
%o A152548 (PARI) a(n)=sum(k=0,floor((n+1)/2),binomial(n+1, k)*(n+1-2*k)^3)/(n+1)
%Y A152548 Cf. A008288, A152547, A107233, A014477.
%K A152548 nonn
%O A152548 0,2
%A A152548 _Paul D. Hanna_, Dec 14 2008

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