A181474 Sequence related to the Hankel transform of A105523(n+5).

1, 2, 10, 15, 42, 56, 120, 150, 275, 330, 546, 637, 980, 1120, 1632, 1836, 2565, 2850, 3850, 4235, 5566, 6072, 7800, 8450, 10647, 11466, 14210, 15225, 18600, 19840, 23936, 25432, 30345, 32130, 37962, 40071, 46930

Offset: 0

Author

Paul Barry, Oct 22 2010

Keywords

• easy
• nonn

Comments

The Hankel transform of A105523(n+5) is (-1)^C(n+2,2)a(n+1).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Nathaniel K. Green and Edward D. Kim, Further techniques on a polynomial positivity question of Collins, Dykema, and Torres-Ayala, arXiv:2307.06311 [math.OC], 2023. See p. 18.
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Programs

Mathematica

CoefficientList[Series[(1 + x + 4 x^2 + x^3 + x^4)/((1 - x)^5 (1 + x)^4), {x, 0, 36}], x] (* Michael De Vlieger, Jul 25 2023 *)

Formula

G.f.: (1+x+4x^2+x^3+x^4)/((1-x)^5(1+x)^4);

a(n) = 2^n*(n+2)*(n+3)*Gamma(floor(n/2)+3)*Gamma(floor((n+1)/2)+1/2)/(12n!*sqrt(Pi)) (suggested by WolframAlpha).

a(n) = +a(n-1) +4*a(n-2) -4*a(n-3) -6*a(n-4) +6*a(n-5) +4*a(n-6) -4*a(n-7) -a(n-8) +a(n-9). a(n) = (n+3)*(n+2)*(2*n^2+2*(-1)^n*n+10*n+5*(-1)^n+11)/96. [R. J. Mathar, Oct 23 2010]