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URL: https://oeis.org/A391219

⇱ A391219 - OEIS

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A391219
Expansion of 1/(g * (2-g))^3, where g = 1+x*g^3 is the g.f. of A001764.
4
1, 0, 3, 18, 105, 618, 3682, 22182, 134910, 827160, 5106486, 31712142, 197953171, 1241244522, 7814173800, 49368550568, 312896022834, 1988826267600, 12674329217998, 80962608693150, 518308671138447, 3324764457535774, 21366568180376802, 137546923635660054
OFFSET
0,3
LINKS
FORMULA
G.f.: B(x)^3, where B(x) is the g.f. of A391208.
G.f.: 1/(1 - x^2*g^6)^3, where g = 1+x*g^3 is the g.f. of A001764.
a(n) = (1/(n-1)) * Sum_{k=0..n} (k-1) * binomial(k+2,2) * binomial(3*n-3,n-k) for n > 1.
a(n) = (6/n) * Sum_{k=0..n-1} binomial(k+2,3) * binomial(3*n-4,n-1-k) for n > 0.
a(n) = (6/n) * Sum_{k=0..floor(n/2)} binomial(k+2,3) * binomial(3*n,n-2*k) for n > 0.
MATHEMATICA
Join[{1}, Table[Sum[(6/n)*Binomial[k+2, 3]*Binomial[3*n, n-2*k], {k, 0, Floor[n/2]}], {n, 1, 25}]] (* Vincenzo Librandi, Dec 04 2025 *)
PROG
(PARI) a(n) = if(n==0, 1, 6/n*sum(k=0, n\2, binomial(k+2, 3)*binomial(3*n, n-2*k)));
(Magma) [1] cat [(6 / n) * &+[Binomial(k+2, 3) * Binomial(3*n, n-2*k): k in [0..Floor(n/2)]] : n in [1..30] ]; // Vincenzo Librandi, Dec 04 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 04 2025
STATUS
approved