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

0,1

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

Links

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

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).

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A033552, A197433, A161227 - A161239.

Sequence in context: A155076 A328308 A257196 * A290808 A364252 A190239

Adjacent sequences: A176134 A176135 A176136 * A176138 A176139 A176140

Keyword

nonn

Author

Reinhard Zumkeller, Apr 09 2010

Status

approved