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Operad: Quadri-algebra

abbreviation
Quad
category
vec
type
non-symetrique
dimensions

1, 4, 23, 156, 1162

OEIS: A007297

Formula: sum(binomial(3*n,n+1+j)*binomial(j-1,j-n) for j in range(n,2*n))/n

Serie: (x*(1-x)/(1+x)**3).reverse()

properties
binary, quadratic, Koszul, cyclic
Koszul dual operad
Quad!
generateurs
  • generator ↖ of arity 2
  • generator ↗ of arity 2
  • generator ↘ of arity 2
  • generator ↙ of arity 2
relations
  • equality: (x↖y)↖z = x↖(y↖z)+x↖(y↗z)+x↖(y↘z)+x↖(y↙z)
  • equality: (x↗y)↖z = x↗(y↖z)+x↗(y↙z)
  • equality: (x↖y)↗z+(x↗y)↗z = x↗(y↗z)+x↗(y↘z)
  • equality: (x↙y)↖z = x↙(y↖z)+x↙(y↗z)
  • equality: (x↘y)↖z = x↘(y↖z)
  • equality: (x↘y)↗z+(x↙y)↗z = x↘(y↗z)
  • equality: (x↖y)↙z+(x↙y)↙z = x↙(y↙z)+x↙(y↘z)
  • equality: (x↗y)↙z+(x↘y)↙z = x↘(y↙z)
  • equality: (x↖y)↘z+(x↗y)↘z+(x↘y)↘z+(x↙y)↘z = x↘(y↘z)

References