A245798 Catalan number analogs for totienomial coefficients (A238453).

1, 1, 2, 4, 12, 36, 120, 360, 960, 3840, 13824, 41472, 152064, 506880, 2280960, 7983360, 29937600, 99792000, 266112000, 1197504000, 4790016000, 19160064000, 73156608000, 219469824000, 1009561190400, 3533464166400, 12563428147200, 54441521971200, 155547205632000

Offset

0,3

Comments

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Euler's totient function (A000010).

When the INTEGERS (2014) paper was written it was not known that this was an integral sequence (see the final paragraph of that paper). However, it is now known to be integral.

Another name could be phi-Catalan numbers. - Tom Edgar, Mar 29 2015

Links

Tom Edgar, Table of n, a(n) for n = 0..28

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Formula

a(n) = A238453(2*n,n) / A000010(n+1).

Example

We see that A238453(10,5) = 72 and A000010(5+1) = 2, so a(5) = (1/2)*72 = 36.

Prog

(Sage)
[(1/euler_phi(n+1))*prod(euler_phi(i) for i in [1..2*n])/prod(euler_phi(i) for i in [1..n])^2 for n in [0..100]]

Crossrefs

Cf. A000010, A238453, A000108, A001088.

Sequence in context: A117757 A009623 A148208 * A241530 A123071 A048116

Adjacent sequences: A245795 A245796 A245797 * A245799 A245800 A245801

Keyword

nonn

Author

Tom Edgar, Aug 22 2014

Status

approved