A309303 Expansion of g.f. (sqrt(x+1) - sqrt(1-3*x))/(2*(x+1)^(3/2)).

0, 1, -1, 2, -1, 4, 2, 13, 23, 68, 164, 439, 1146, 3067, 8231, 22306, 60791, 166684, 459308, 1271479, 3534116, 9859573, 27598757, 77490472, 218183522, 615902899, 1742738477, 4942022648, 14043034703, 39979680748, 114020882010, 325721340061

Offset: 0

Author

Vladimir Reshetnikov, Jul 21 2019

Keywords

• sign

Comments

(-1)^a(n) = (-1)^A010060(n) = A106400(n) (Thue-Morse sequence).

a(n) + a(n+1) = A005043(n) = (-1)^n * hypergeom([-n, 1/2], [2], 4) (Motzkin sums).

Links

• Robert Israel, Table of n, a(n) for n = 0..2106
• Eric Weisstein's World of Mathematics, Thue-Morse sequence.
• Wikipedia, Thue-Morse sequence.

Crossrefs

Cf. A005043, A010060, A106400.

Programs

Maple

f:= gfun:-rectoproc({n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)),a(0)=0,a(1)=1,a(2)=-1},a(n),remember):
map(f, [$0..40]); # Robert Israel, Jul 23 2019

Mathematica

Table[(-1)^n/2 + 3^(n + 3/2)/2^(n + 4) (2 n - 3)!!/n! Hypergeometric2F1[3/2, 3/2, 3/2 - n, 1/4], {n, 0, 31}]

Formula

a(n) = (-1)^n/2 + 3^(n+3/2)/2^(n+4) * (2*n-3)!!/n! * hypergeom([3/2, 3/2], [3/2 - n], 1/4).

D-finite with recurrence: n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)).

a(n) ~ c * 3^n / n^(3/2), where c = 3^(3/2) / (32*sqrt(Pi)) = 0.09161297...