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A197433 Sum of distinct Catalan numbers: a(n) = Sum_{k>=0} A030308(n,k)*C(k+1) where C(n) is the n-th Catalan number (A000108). (C(0) and C(1) not treated as distinct.)

%I A197433 #37 Nov 18 2021 03:59:37
%S A197433 0,1,2,3,5,6,7,8,14,15,16,17,19,20,21,22,42,43,44,45,47,48,49,50,56,
%T A197433 57,58,59,61,62,63,64,132,133,134,135,137,138,139,140,146,147,148,149,
%U A197433 151,152,153,154,174,175,176,177,179,180,181,182,188,189,190,191,193,194,195,196
%N A197433 Sum of distinct Catalan numbers: a(n) = Sum_{k>=0} A030308(n,k)*C(k+1) where C(n) is the n-th Catalan number (A000108). (C(0) and C(1) not treated as distinct.)
%C A197433 Replace 2^k with A000108(k+1) in binary expansion of n.
%C A197433 From _Antti Karttunen_, Jun 22 2014: (Start)
%C A197433 On the other hand, A244158 is similar, but replaces 10^k with A000108(k+1) in decimal expansion of n.
%C A197433 This sequence gives all k such that A014418(k) = A239903(k), which are precisely all nonnegative integers k whose representations in those two number systems contain no digits larger than 1. From this also follows that this is a subsequence of A244155.
%C A197433 (End)
%H A197433 Antti Karttunen, <a href="/A197433/b197433.txt">Table of n, a(n) for n = 0..8191</a>
%F A197433 For all n, A244230(a(n)) = n. - _Antti Karttunen_, Jul 18 2014
%F A197433 G.f.: (1/(1 - x))*Sum_{k>=0} Catalan number(k+1)*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jul 23 2017
%t A197433 nmax = 63;
%t A197433 a[n_] := If[n == 0, 0, SeriesCoefficient[(1/(1-x))*Sum[CatalanNumber[k+1]* x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, n] // Ceiling}], {x, 0, n}]];
%t A197433 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 18 2021, after _Ilya Gutkovskiy_ *)
%Y A197433 Characteristic function: A176137.
%Y A197433 Subsequence of A244155.
%Y A197433 Cf. A000108, A030308, A197432, A014418, A239903, A244158, A244159, A244230, A244231, A244232, A244315, A244316.
%Y A197433 Cf. also A060112.
%Y A197433 Other sequences that are built by replacing 2^k in binary representation with other numbers: A022290 (Fibonacci), A029931 (natural numbers), A059590 (factorials), A089625 (primes), A197354 (odd numbers).
%K A197433 easy,nonn
%O A197433 0,3
%A A197433 _Philippe Deléham_, Oct 15 2011
%E A197433 Name clarified by _Antti Karttunen_, Jul 18 2014