A328002 a(n) = 2^n * Sum_{k=0..n} Product_{j=1..k} (2/j)^((-1)^j).

1, 3, 8, 22, 50, 130, 280, 700, 1470, 3570, 7392, 17556, 36036, 84084, 171600, 394680, 802230, 1823250, 3695120, 8314020, 16812796, 37505468, 75716368, 167657672, 338019500, 743642900, 1497686400, 3276189000, 6592494600, 14348370600, 28851858720, 62512360560

Offset: 0

Author

Peter Luschny, Oct 02 2019

Keywords

• nonn

Links

• Manuel Kauers, Algorithms for D-finite functions.
• Wikipedia, Holonomic_function

Crossrefs

Cf. A056040, row sums of A338654.

Programs

Maple

s := z -> 2^z*(GAMMA(z/2+1/2)/GAMMA(z/2+1))^cos(z*Pi):
a := n -> sqrt(2/Pi)*add(s(k)*s(n-k-1/2), k=0..n): seq(a(n), n=0..19);
# Alternative:
a := proc(n) option remember; if n=0 then 1 elif n=1 then 3 else
(4*n*(n*(n+5)-10)*a(n-2) + 2*(n-6)*(n+2)*a(n-1))/(n*(n*(n+3)-14)) fi end:
seq(a(n), n=0..31); # Peter Luschny, Jan 10 2020

Formula

a(n) = sqrt(2/Pi)*Sum_{k=0..n} s(k)*s(n-k-1/2) where s(n) = 2^n*(Gamma(n/2 + 1/2)/Gamma(n/2 + 1))^cos(n*Pi).

a(n) = [x^n] (4*x^2 - x - 1)/((1 - 4*x^2)^(3/2)*(2*x - 1)).

a(n) ~ 2^n * n^(3/2) / (3*sqrt(2*Pi)). - Vaclav Kotesovec, Oct 19 2019

D-finite with recurrence: n*a(n) +(n-4)*a(n-1) +2*(-4*n+3)*a(n-2) +4*(-n+2)*a(n-3) +16*(n-2)*a(n-4)=0. - R. J. Mathar, Jan 09 2020

D-finite with recurrence: a(n) = (4*n*(n*(n + 5) - 10)*a(n-2) + 2*(n - 6)*(n + 2)*a(n-1))/(n*(n*(n + 3) - 14)). - Peter Luschny, Jan 10 2020

Let q(n) = a(n+1)/a(n) and d(n) = numerator(q(n)) - 2*denominator(q(n)).

Conjecture: d(n) is periodic with period 12 and repetition pattern (1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 3, 1). - Peter Luschny, Jan 10 2020

a(n) = Sum_{k=0..n} 2^(n-k)*A056040(k). - Peter Luschny, Apr 22 2021

Extensions

Simpler name by Peter Luschny, Apr 22 2021