A393246 a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,k) * binomial(n-k,3*k).

1, 1, 1, 1, 3, 9, 21, 41, 77, 155, 337, 745, 1611, 3413, 7217, 15425, 33299, 72101, 155879, 336539, 727175, 1574665, 3416533, 7420657, 16125397, 35057795, 76271285, 166069349, 361859271, 788952801, 1721005769, 3755971929, 8201097081, 17915415387, 39153825545

Offset

0,5

Links

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

Formula

G.f.: 1/sqrt((1-x)^2 - 4*x^4/(1-x)).

D-finite with recurrence: (6 + 4*n)*a(n) + (-10 - 3*n)*a(n + 1) + (-11 - 4*n)*a(n + 2) + (21 + 6*n)*a(n + 3) + (-17 - 4*n)*a(n + 4) + (n + 5)*a(n + 5) = 0. - Robert Israel, Feb 09 2026

Maple

f:= gfun:-rectoproc({(6 + 4*n)*a(n) + (-10 - 3*n)*a(n + 1) + (-11 - 4*n)*a(n + 2) + (21 + 6*n)*a(n + 3) + (-17 - 4*n)*a(n + 4) + (n + 5)*a(n + 5), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 3}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 09 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-k, 3*k], {k, 0, Floor[n/4]}], {n, 0, 31}] (* Vincenzo Librandi, Feb 08 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(2*k, k)*binomial(n-k, 3*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-k, 3*k) : k in [0..Floor(n/4)]] : n in [0..35] ]; // Vincenzo Librandi, Feb 08 2026

Crossrefs

Cf. A098482, A383355, A393245.

Sequence in context: A024173 A396151 A097119 * A374585 A146608 A161214

Adjacent sequences: A393243 A393244 A393245 * A393247 A393248 A393249

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved