OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: g/((1-3*x*g^2) * (1-x*g^5)) where g = 1+x*g^3 is the g.f. of A001764.
From Vaclav Kotesovec, Nov 09 2025: (Start)
Recurrence: 2*n*(2*n - 3)*(37*n^4 - 910*n^3 + 5747*n^2 - 13874*n + 11520)*a(n) = 3*(629*n^6 - 16580*n^5 + 130155*n^4 - 460480*n^3 + 808396*n^2 - 671280*n + 201600)*a(n-1) - 2*(2627*n^6 - 70049*n^5 + 588211*n^4 - 2299771*n^3 + 4624422*n^2 - 4624560*n + 1814400)*a(n-2) - (4847*n^6 - 128978*n^5 + 1070437*n^4 - 4117882*n^3 + 8116896*n^2 - 7926480*n + 3024000)*a(n-3) - 3*(3*n - 10)*(3*n - 8)*(37*n^4 - 762*n^3 + 3239*n^2 - 4962*n + 2520)*a(n-4).
a(n) ~ (31 + ((50933 - 4371*sqrt(93))/2)^(1/3) + (31*(1643 + 141*sqrt(93))/2)^(1/3)) * (2 + ((729 - 3*sqrt(93))^(1/3)/3 + (243 + sqrt(93))^(1/3)/3^(2/3))/2^(1/3))^n / 93 - 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n-1)). (End)
MATHEMATICA
Table[Sum[Binomial[3*n+2*k+1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Nov 08 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(3*n+2*k+1, n-k));
(Magma) [&+[Binomial(3*n+2*k+1, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 08 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 06 2025
STATUS
approved
