A390237 Expansion of g/(1 - x*g^2)^2, where g = 1+x*g^4 is the g.f. of A002293.

1, 3, 13, 71, 445, 3040, 21998, 165801, 1287993, 10240870, 82939364, 681829854, 5674948844, 47726973679, 404963281675, 3462485634943, 29802561951657, 258025475286759, 2245560767095261, 19633470022380120, 172374674900586262, 1519071406103018848

Offset

0,2

Links

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

Robert Israel, Recurrence of order 8

Formula

a(n) = Sum_{k=0..n} (k+1) * (2*k+1) * binomial(4*n-2*k+1,n-k)/(4*n-2*k+1).

D-finite with recurrence of order 8 (see link). - Robert Israel, Jul 23 2026

Maple

with(gfun):
e1:= -g+1+x*g^4:
e2:= z*(1-x*g^2)^2-g:
eq:= factor(resultant(e1, e2, g)):
de:= algeqtodiffeq(eq/x^3, z(x)):
rec:= diffeqtorec(de, z(x), a(n)):
f:= rectoproc(rec, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 23 2026

Mathematica

Table[Sum[(k+1)*(2*k+1)*Binomial[4*n-2*k+1, n-k]/(4*n-2*k+1), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*(2*k+1)*binomial(4*n-2*k+1, n-k)/(4*n-2*k+1));
(Magma) [&+[(k+1)*(2*k+1)*Binomial(4*n-2*k+1, n-k)/(4*n-2*k+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 03 2025

Crossrefs

Cf. A002293, A390709, A391198.

Sequence in context: A198447 A318223 A162326 * A122455 A126390 A272428

Adjacent sequences: A390234 A390235 A390236 * A390238 A390239 A390240

Keyword

nonn,changed

Author

Seiichi Manyama, Dec 02 2025

Status

approved