OFFSET
0,2
COMMENTS
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.
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
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
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
FORMULA
a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+2, 2)*(1+(-1)^(n-k))/(n+k+2).
a(n) = 2^n + Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n, k). - Nathaniel Johnston, Apr 20 2011
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
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
a(n) ~ 2^(n+1/2)*sqrt(n)/sqrt(Pi) * (1 + sqrt(2*Pi/n)/4). - Vaclav Kotesovec, Feb 13 2014
MAPLE
a:=proc(n)return 2^n+add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end:
seq(a(n), n=0..30); # Nathaniel Johnston, Apr 20 2011
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 12 2004
STATUS
approved