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.)

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

Offset

0,3

Comments

Replace 2^k with A000108(k+1) in binary expansion of n.

From Antti Karttunen, Jun 22 2014: (Start)

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

Antti Karttunen, Table of n, a(n) for n = 0..8191

Formula

For all n, A244230(a(n)) = n. - Antti Karttunen, Jul 18 2014

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/(1-x))*Sum[CatalanNumber[k+1]* x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, n] // Ceiling}], {x, 0, n}]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy *)

Crossrefs

Characteristic function: A176137.

Subsequence of A244155.

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

Cf. also A060112.

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).

Sequence in context: A287339 A039086 A280060 * A153781 A255398 A050025

Adjacent sequences: A197430 A197431 A197432 * A197434 A197435 A197436

Keyword

easy,nonn

Author

Philippe Deléham, Oct 15 2011

Extensions

Name clarified by Antti Karttunen, Jul 18 2014

Status

approved