OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: g^2/((1-3*x*g^2) * (1-x*g)) where g = 1+x*g^3 is the g.f. of A001764.
From Vaclav Kotesovec, Nov 09 2025: (Start)
a(n) = Sum_{k=0..n} binomial(n+2*k+2, k).
a(n) ~ 3^(3*n + 9/2) / (7*sqrt(Pi*n) * 2^(2*n+3)). (End)
From Seiichi Manyama, Nov 10 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n+k+4,n-k).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-k+3,n-2*k). (End)
D-finite with recurrence: (-540*n^2 - 2160*n - 2100)*a(n) + (1025*n^2 + 5355*n + 7060)*a(n + 1) + (-1355*n^2 - 8505*n - 14500)*a(n + 2) + (882*n^2 + 7116*n + 14620)*a(n + 3) + (-401*n^2 - 3975*n - 9842)*a(n + 4) + (71*n^2 + 825*n + 2402)*a(n + 5) + (-4*n^2 - 54*n - 182)*a(n + 6) = 0. - Robert Israel, Jul 23 2026
MAPLE
f:= gfun:-rectoproc({(-540*n^2 - 2160*n - 2100)*a(n) + (1025*n^2 + 5355*n + 7060)*a(n + 1) + (-1355*n^2 - 8505*n - 14500)*a(n + 2) + (882*n^2 + 7116*n + 14620)*a(n + 3) + (-401*n^2 - 3975*n - 9842)*a(n + 4) + (71*n^2 + 825*n + 2402)*a(n + 5) + (-4*n^2 - 54*n - 182)*a(n + 6), a(0) = 1, a(1) = 6, a(2) = 35, a(3) = 209, a(4) = 1275, a(5) = 7904}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 23 2026
MATHEMATICA
a[n_]:=Sum[Binomial[3*n-2*k+2, n-k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Vincenzo Librandi, Oct 31 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(3*n-2*k+2, n-k));
(Magma) [&+[Binomial(3*n-2*k+2, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Seiichi Manyama, Oct 30 2025
STATUS
approved