

A197433


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


13



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, 57, 58, 59, 61, 62, 63, 64, 132, 133, 134, 135, 137, 138, 139, 140, 146, 147, 148, 149, 151, 152, 153, 154, 174, 175, 176, 177, 179, 180, 181, 182, 188, 189, 190, 191, 193, 194, 195, 196
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OFFSET

0,3


COMMENTS

Replace 2^k with A000108(k+1) in binary expansion of n.
On the other hand, A244158 is similar, but replaces 10^k with A000108(k+1) in decimal expansion of n.
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.
(End)


LINKS



FORMULA

G.f.: (1/(1  x))*Sum_{k>=0} Catalan number(k+1)*x^(2^k)/(1 + x^(2^k)).  Ilya Gutkovskiy, Jul 23 2017


MATHEMATICA

nmax = 63;
a[n_] := If[n == 0, 0, SeriesCoefficient[(1/(1x))*Sum[CatalanNumber[k+1]* x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, n] // Ceiling}], {x, 0, n}]];


CROSSREFS

Cf. A000108, A030308, A197432, A014418, A239903, A244158, A244159, A244230, A244231, A244232, A244315, A244316.
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).


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



