A176137 Number of partitions of n into distinct Catalan numbers, cf. A000108.

1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Reinhard Zumkeller, Apr 09 2010

Keywords

• nonn

Comments

a(n) <= 1;

a(A000108(n)) = 1; a(A141351(n)) = 1; a(A014138(n)) = 1.

A197433 gives all such numbers k that a(k) = 1, in other words, this is the characteristic function of A197433, and all three sequences mentioned above are its subsequences. - Antti Karttunen, Jun 25 2014

Examples

			56 = 42+14 = A000108(5)+A000108(4), all other sums of distinct Catalan numbers are not equal 56, therefore a(56)=1.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

When right-shifted (prepended with 1) this sequence is the first differences of A244230.

Cf. A033552, A197433, A161227 - A161239.

Programs

Mathematica

nmax = 104;
A197433 = CoefficientList[(1/(1 - x))*Sum[ CatalanNumber[k + 1]*x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, nmax] // Ceiling}] + O[x]^nmax, x];
a[n_] := Boole[MemberQ[A197433, n]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy in A197433 *)

Scheme

(define (A176137 n) (if (zero? n) 1 (- (A244230 (+ n 1)) (A244230 n)))) ;; Antti Karttunen, Jun 25 2014

Formula

a(n) = f(n,1,1) with f(m,k,c) = if c>m then 0^m else f(m-c,k+1,c') + f(m,k+1,c') where c'=2*c*(2*k+1)/(k+2).