A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040

Offset

0,3

Links

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

Formula

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

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

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

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

G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).

G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.

G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.

G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

Mathematica

Table[Sum[(-3)^(n-k)*Binomial[2*n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 31 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));
(Magma) [&+[(-3)^(n-k) * Binomial(2*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Crossrefs

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.

Cf. A226733, A226705, A226751.

Cf. A005810, A079589, A183160.

Sequence in context: A221842 A100214 A229535 * A309618 A047710 A370573

Adjacent sequences: A387082 A387083 A387084 * A387086 A387087 A387088

Keyword

nonn

Author

Seiichi Manyama, Aug 16 2025

Status

approved