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.
D-finite with recurrence (-648*n^2 - 1296*n - 576)*a(n) + (4146*n^2 + 10770*n + 6804)*a(n + 1) + (-9510*n^2 - 45438*n - 56412)*a(n + 2) + (19194*n^2 + 144438*n + 272376)*a(n + 3) + (-26327*n^2 - 251762*n - 600348)*a(n + 4) + (19195*n^2 + 210100*n + 574272)*a(n + 5) + (-7238*n^2 - 87836*n - 266112)*a(n + 6) + (1295*n^2 + 17318*n + 57624)*a(n + 7) + (-84*n^2 - 1218*n - 4368)*a(n + 8) = 0. - Robert Israel, Nov 10 2025
From Vaclav Kotesovec, Nov 18 2025: (Start)
Recurrence (of order 4): 2*n*(2*n - 3)*(595*n^4 - 5974*n^3 + 22169*n^2 - 36110*n + 21840)*a(n) = (25585*n^6 - 303292*n^5 + 1434167*n^4 - 3436208*n^3 + 4345068*n^2 - 2687760*n + 604800)*a(n-1) - 2*(38080*n^6 - 455521*n^5 + 2183474*n^4 - 5346419*n^3 + 7018026*n^2 - 4654800*n + 1209600)*a(n-2) + (82705*n^6 - 994606*n^5 + 4804955*n^4 - 11909414*n^3 + 15948960*n^2 - 10949040*n + 3024000)*a(n-3) - 3*(3*n - 10)*(3*n - 8)*(595*n^4 - 3594*n^3 + 7817*n^2 - 7314*n + 2520)*a(n-4).
a(n) ~ 3^(3*n + 3/2) / (17 * sqrt(Pi*n) * 2^(2*n-1)). (End)
MAPLE
f:= proc(n) local k; add((-1)^k*binomial(3*n+2*k+1, n-k), k=0..n) end proc:
map(f, [$0..30]); # Robert Israel, Nov 10 2025
MATHEMATICA
Table[Sum[(-1)^k*Binomial[3*n+2*k+1, n-k], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Nov 10 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*n+2*k+1, n-k));
(Magma) [&+[(-1)^k*Binomial(3*n+2*k+1, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 10 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 09 2025
STATUS
approved