A385250 a(n) = Sum_{k=0..n} binomial(n+2*k+1,k).

1, 5, 27, 155, 924, 5643, 35036, 220133, 1395549, 8909120, 57193916, 368849781, 2387828041, 15507998588, 100996781823, 659323123227, 4313184213360, 28268290229481, 185572527135506, 1220015370938720, 8031387080176074, 52934002548821884, 349261397277490615

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1201

Formula

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

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

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

Maple

f:= proc(n) local k; add(binomial(n+2*k+1, k), k=0..n) end proc:
map(f, [$0..25]); # Robert Israel, Nov 06 2025

Mathematica

Table[Sum[Binomial[ n+2*k+1, k], {k, 0, n}], {n, 0, 28}] (* Vincenzo Librandi, Jan 04 2026 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+2*k+1, k));
(Magma) [&+[Binomial(n+2*k+1, k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Jan 04 2026

Crossrefs

Cf. A025174, A038744, A160906, A263134, A386839, A390452.

Cf. A371785, A388043, A390239.

Cf. A001764, A390412.

Sequence in context: A332598 A305573 A184702 * A083326 A083880 A363185

Adjacent sequences: A385247 A385248 A385249 * A385251 A385252 A385253

Keyword

nonn

Author

Seiichi Manyama, Nov 06 2025

Status

approved