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URL: https://oeis.org/A390102

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A390102
G.f. A(x) satisfies A(x) = 1 + x/(1-x^2)^3 * A(x)^2.
4
1, 1, 2, 8, 26, 93, 342, 1294, 5010, 19751, 79036, 320215, 1310986, 5415416, 22543108, 94474620, 398275738, 1687825205, 7186221246, 30725239502, 131866322768, 567889032437, 2453304252942, 10628744342355, 46169328098970, 201037260539630, 877347171346140, 3836785348958714, 16811298676724492
OFFSET
0,3
LINKS
FORMULA
G.f.: 2/(1 + sqrt(1 - 4*x/(1-x^2)^3)).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k-1,k) * Catalan(n-2*k).
D-finite with recurrence: (n - 5)*a(n) + (6 - 4*n)*a(n + 2) + (12 + 6*n)*a(n + 4) + (10 + 4*n)*a(n + 5) + (-22 - 4*n)*a(n + 6) + (-30 - 4*n)*a(n + 7) + (9 + n)*a(n + 8) = 0. - Robert Israel, Feb 15 2026
MAPLE
f:= gfun:-rectoproc({(n - 5)*a(n) + (6 - 4*n)*a(n + 2) + (12 + 6*n)*a(n + 4) + (10 + 4*n)*a(n + 5) + (-22 - 4*n)*a(n + 6) + (-30 - 4*n)*a(n + 7) + (9 + n)*a(n + 8), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 8, a(4) = 26, a(5) = 93, a(6) = 342, a(7) = 1294, a(8) = 5010}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 15 2026
MATHEMATICA
Table[Sum[ Binomial[3*n-5*k-1, k]*CatalanNumber[n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)
PROG
(PARI) a(n) = sum(k=0, n\2, binomial(3*n-5*k-1, k)*binomial(2*(n-2*k), n-2*k)/(n-2*k+1));
(Magma) [&+[Catalan(n-2*k) * Binomial(3*n-5*k-1, k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025
CROSSREFS
Cf. A000108.
Sequence in context: A389872 A026638 A307401 * A067855 A301699 A129368
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 25 2025
STATUS
approved