A390680 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k-3,n-2*k).

1, -1, -1, 0, 3, 14, 58, 234, 931, 3674, 14434, 56576, 221518, 867008, 3393544, 13286350, 52040483, 203936538, 799621994, 3137022800, 12313867738, 48362815668, 190047775804, 747206147380, 2939246350478, 11567527538964, 45545462566148, 179407603771704

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n+k-1,n-k).

From Robert Israel, Nov 18 2025: (Start)

a(n) = binomial(2*n - 1, n) * hypergeom([1, -n, 2*n], [n/2, n/2 + 1/2], 1/2).

D-finite with recurrence (20*n^3 + 46*n^2 + 42*n + 12)*a(n) + (-45*n^3 - 106*n^2 - 83*n - 30)*a(n + 1) + (50*n^3 + 120*n^2 + 82*n + 36)*a(n + 2) + (-10*n^3 - 28*n^2 + 2*n - 12)*a(n + 3) = 0 for n >= 1. (End)

Maple

seq(simplify(binomial(2*n - 1, n) * hypergeom([1, -n, 2*n], [n/2, n/2 + 1/2], 1/2)), n=0..30); # Robert Israel, Nov 18 2025

Mathematica

Table[Sum[(-2)^k*Binomial[2*n+k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Nov 18 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k-3, n-2*k));
(Magma) [&+[(-2)^k*Binomial(2*n+k-1, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 18 2025

Crossrefs

Cf. A176332, A371798.

Cf. A390681, A390682.

Sequence in context: A127363 A133444 A126875 * A110526 A319857 A038679

Adjacent sequences: A390677 A390678 A390679 * A390681 A390682 A390683

Keyword

sign

Author

Seiichi Manyama, Nov 15 2025

Status

approved