A391084 Expansion of g^2/(1 - x^2*g^6), where g = 1+x*g^4 is the g.f. of A002293.

1, 2, 10, 60, 401, 2864, 21400, 165220, 1307620, 10552302, 86499898, 718258092, 6028798270, 51069425360, 436027711460, 3748369529060, 32417585857772, 281855861501960, 2462233783684360, 21601019155153160, 190230596956328785, 1681092794230175680

Offset

0,2

Links

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

Formula

G.f.: g^3/(1 + x*g^3), where g = 1+x*g^4 is the g.f. of A002293.

G.f.: B(x)/(x * (1 + B(x))), where B(x) is the g.f. of A006632.

a(n) = Sum_{k=0..floor(n/2)} (3*k+1) * binomial(4*n-2*k+2,n-2*k)/(2*n-k+1).

a(n) = (1/(3*n+2)) * Sum_{k=0..floor(n/2)} (6*k+2) * binomial(4*n-2*k+1,n-2*k).

a(n) = Sum_{k=0..n} (-1)^k * (3*k+3) * binomial(4*n-k+3,n-k)/(4*n-k+3).

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * (k+1) * binomial(4*n-k+2,n-k).

Mathematica

Table[Sum[ (3*k+1)*Binomial[4*n-2*k+2, n-2*k]/(2*n-k+1), {k, 0, Floor[n/2]}], {n, 0, 26}] (* Vincenzo Librandi, Nov 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (3*k+1)*binomial(4*n-2*k+2, n-2*k)/(2*n-k+1));
(Magma) [&+[(3*k+1)*Binomial(4*n-2*k+2, n-2*k)/(2*n-k+1): k in [0..Floor(n/2)]] : n in [0..40] ]; // Vincenzo Librandi, Nov 30 2025

Crossrefs

Cf. A389112, A390147, A391080, A391082.

Cf. A002293, A006632.

Sequence in context: A215002 A301625 A262001 * A276310 A372578 A098616

Adjacent sequences: A391081 A391082 A391083 * A391085 A391086 A391087

Keyword

nonn

Author

Seiichi Manyama, Nov 27 2025

Status

approved