A391129 Expansion of g^2/(1 - x^2*g^4), where g = 1+x*g^3 is the g.f. of A001764.

1, 2, 8, 36, 177, 920, 4972, 27656, 157283, 910378, 5345408, 31761232, 190616428, 1153815776, 7035931524, 43182501648, 266538732951, 1653488312726, 10303782450616, 64468708569236, 404844870033361, 2550760645474016, 16120015360974736, 102156302157189536

Offset

0,2

Links

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

Formula

G.f.: g^3/(1 + x*g^2), where g = 1+x*g^3 is the g.f. of A001764.

a(n) = Sum_{k=0..floor(n/2)} (4*k+2) * binomial(3*n-2*k+2,n-2*k)/(3*n-2*k+2).

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

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

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

Mathematica

Table[Sum[ (4*k+2)*Binomial[3*n -2*k+2, n-2*k]/(3*n-2*k+2), {k, 0, Floor[n/2]}], {n, 0, 26}] (* Vincenzo Librandi, Nov 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (4*k+2)*binomial(3*n-2*k+2, n-2*k)/(3*n-2*k+2));
(Magma) [&+[(4*k+2)*Binomial(3*n-2*k+2, n-2*k)/(3*n-2*k+2): k in [0..Floor(n/2)]] : n in [0..40] ]; // Vincenzo Librandi, Nov 30 2025

Crossrefs

Cf. A109972, A391122, A391126.

Cf. A089354, A121545.

Cf. A001764, A006629.

Sequence in context: A109318 A113327 A227791 * A245102 A129148 A285672

Adjacent sequences: A391126 A391127 A391128 * A391130 A391131 A391132

Keyword

nonn

Author

Seiichi Manyama, Nov 29 2025

Status

approved