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URL: https://oeis.org/A367027

⇱ A367027 - OEIS


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A367027
G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^5.
2
1, 1, 2, 4, 5, -13, -147, -816, -3534, -12650, -35420, -53040, 199056, 2391340, 14555740, 68264112, 261045693, 769660569, 1167906402, -5145668100, -61758940705, -385813067255, -1857144860445, -7266981925560, -21793022441775, -32643056947527, 161919845140752
OFFSET
0,3
LINKS
FORMULA
a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-k,k) * binomial(3*n-2*k,n-2*k).
G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^2 ) )^(1/2). - Seiichi Manyama, Mar 08 2025
D-finite with recurrence: -50*(5*n + 6)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 5*(3041*n^4 + 15872*n^3 + 32443*n^2 + 31300*n + 12096)*a(n + 1) - 4*(2*n + 5)*(n + 2)*(331*n^2 + 1025*n + 1044)*a(n + 2) + 36*(2*n + 5)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 3) = 0. - Robert Israel, Mar 08 2026
MAPLE
f:= gfun:-rectoproc({-50*(5*n + 6)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 5*(3041*n^4 + 15872*n^3 + 32443*n^2 + 31300*n + 12096)*a(n + 1) - 4*(2*n + 5)*(n + 2)*(331*n^2 + 1025*n + 1044)*a(n + 2) + 36*(2*n + 5)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 3), a(0) = 1, a(1) = 1, a(2) = 2}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 08 2026
PROG
(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-k, k)*binomial(3*n-2*k, n-2*k))/(2*n+1);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 02 2023
STATUS
approved