%I #30 Jul 02 2021 16:46:26
%S 1,3,7,16,35,76,162,344,723,1516,3158,6568,13598,28120,57956,119344,
%T 245123,503116,1030542,2109704,4311786,8808328,17969372,36644176,
%U 74640430,151985016,309170332,628741264,1277540828,2595198256
%N Expansion of (sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(3/2)).
%C The g.f. is transformed to 1/(1-x)^3 under the Chebyshev transformation A(x)->1/(1+x^2)A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x), where c(x) is the g.f. of the Catalan numbers A000108.
%C Image of 2n+1 under the Riordan array (1/sqrt(1-4x^2),xc(x^2)). Hankel transform is (n+1)*(-1)^n. - _Paul Barry_, Oct 06 2007
%C a(n) is the minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n+1, and each number in a higher row is the sum of the two numbers directly below it. - _Nathaniel Johnston_, Apr 20 2011
%H Nathaniel Johnston, <a href="/A099325/b099325.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%F a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+2, 2)*(1+(-1)^(n-k))/(n+k+2).
%F a(n) = 2^n + Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n, k). - _Nathaniel Johnston_, Apr 20 2011
%F a(n) = M^n*V topmost term. M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals, [1,0,0,0,...] as the main diagonal; and the rest zeros. V = the vector [1,2,3,...]. - _Gary W. Adamson_, Jan 30 2012
%F D-finite with recurrence n*a(n) + 2*(-n-1)*a(n-1) + 4*(-n+3)*a(n-2) + 8*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Nov 30 2012
%F a(n) ~ 2^(n+1/2)*sqrt(n)/sqrt(Pi) * (1 + sqrt(2*Pi/n)/4). - _Vaclav Kotesovec_, Feb 13 2014
%p a:=proc(n)return 2^n+add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end:
%p seq(a(n), n=0..30); # _Nathaniel Johnston_, Apr 20 2011
%t CoefficientList[Series[(Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(3/2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y Cf. A066411, A099326, A099327.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 12 2004