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%I #12 Nov 18 2021 12:18:38
%S 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,
%T 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,
%U 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
%N Number of partitions of n into distinct Catalan numbers, cf. A000108.
%C a(n) <= 1;
%C a(A000108(n)) = 1; a(A141351(n)) = 1; a(A014138(n)) = 1.
%C 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
%H Reinhard Zumkeller, <a href="/A176137/b176137.txt">Table of n, a(n) for n = 0..10000</a>
%F 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).
%e 56 = 42+14 = A000108(5)+A000108(4), all other sums of distinct Catalan numbers are not equal 56, therefore a(56)=1.
%t nmax = 104;
%t 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];
%t a[n_] := Boole[MemberQ[A197433, n]];
%t Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 18 2021, after _Ilya Gutkovskiy_ in A197433 *)
%o (Scheme) (define (A176137 n) (if (zero? n) 1 (- (A244230 (+ n 1)) (A244230 n)))) ;; _Antti Karttunen_, Jun 25 2014
%Y When right-shifted (prepended with 1) this sequence is the first differences of A244230.
%Y Cf. A033552, A197433, A161227 - A161239.
%K nonn
%O 0,1
%A _Reinhard Zumkeller_, Apr 09 2010
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