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A393245

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

1, 1, 1, 1, 3, 11, 31, 71, 147, 307, 691, 1651, 3981, 9413, 21841, 50441, 117331, 275571, 650659, 1536835, 3624313, 8541233, 20147293, 47608437, 112678845, 266940861, 632697261, 1500171661, 3558812323, 8447803531, 20066108911, 47690196951, 113396798291, 269744856019

(list; graph; refs; listen; history; text; internal format)

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)^2).

D-finite with recurrence: (3 + 3*n)*a(n) + (-3 + n)*a(n + 1) + (-26 - 10*n)*a(n + 2) + (34 + 10*n)*a(n + 3) + (-21 - 5*n)*a(n + 4) + (n + 5)*a(n + 5) = 0. - Robert Israel, Feb 09 2026

a(n) ~ (1 + sqrt(2))^(n + 1/2) / (2^(5/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 10 2026

MAPLE

f:= gfun:-rectoproc({(3 + 3*n)*a(n) + (-3 + n)*a(n + 1) + (-26 - 10*n)*a(n + 2) + (34 + 10*n)*a(n + 3) + (-21 - 5*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, 4*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, 4*k));

(Magma) [&+[Binomial(2*k, k)* Binomial(n, 4*k) : k in [0..Floor(n/4)]] : n in [0..35] ]; // Vincenzo Librandi, Feb 08 2026

(Python)

from math import comb

def A393245(n): return sum(comb(2*k, k) * comb(n, 4*k) for k in range(n // 4 + 1)) # Aitzaz Imtiaz, Feb 09 2026

CROSSREFS

Cf. A098482, A383355, A393246.

Cf. A026375, A002426, A393244.