A094389 -id:A094389 - cp's OEIS frontend

A038003 Odd Catalan numbers: a(n) = A000108(2^n-1).

1, 1, 5, 429, 9694845, 14544636039226909, 94295850558771979787935384946380125, 11311095732253345760960290897769189975961199415637572612957718759342193629

Offset: 0

Christian G. Bower

nonn

The next term has 150 digits. - Harvey P. Dale, Feb 22 2016

David Wasserman, May 07 2007, Table of n, a(n) for n = 0..9
H-Y. Lin, Odd Catalan Numbers modulo 2^k, Integers 11 (2011) #A55
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A094389, A119861, A119908, A120274, A120275.

Intersection of A001790 and A098597. - Dimitri Papadopoulos, Oct 28 2016

[Binomial(2^(n+1)-2, 2^n-1)/(2^n): n in [0..10]]; // Vincenzo Librandi, Nov 01 2016

Select[CatalanNumber[Range[0,300]],OddQ] (* Harvey P. Dale, Feb 22 2016 *)

a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n); \\ Joerg Arndt, Nov 05 2015

from _future_ import division
A038003_list, c, s = [1, 1], 1, 3
for n in range(2,10**5+1):
    c = (c*(4*n-2))//(n+1)
    if n == s:
        A038003_list.append(c)
        s = 2*s+1 # Chai Wah Wu, Feb 12 2015

a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n).
a(n-1) = C(2^n,2^(n-1))/(2^n - 1)/2. - Benoit Cloitre, Aug 17 2002
a(n) = A000108(2^n-1). - David Wasserman, May 07 2007

A119861 Number of distinct prime factors of the odd Catalan numbers A038003(n).

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 36, 64, 117, 209, 381, 699, 1291, 2387, 4445, 8317, 15645, 29494, 55855, 106021, 201778, 384941, 735909, 1409683, 2705277, 5200202

Offset: 1

Alexander Adamchuk, Jul 31 2006, Oct 11 2007

nonn

A038003[n] = A000108[2^n-1] = binomial(2^(n+1)-2, 2^n-1)/(2^n). a(1) = 0 because A038003[1] = 1. a(2) = 1 because A038003[2] = 5. a(3) = 3 because A038003[3] = 429 = 3*11*13. a(4) = 6 because A038003[4] = 9694845 = 3^2*5*17*19*23*29.

Odd Catalan numbers are listed in A038003[n] = A000108[2^n-1] = binomial(2^(n+1)-2, 2^n-1)/(2^n).

Eric Weisstein's World of Mathematics, Catalan Number.

Cf. A038003, A000108, A120274, A120275.

Cf. A000108 = Catalan Number. Cf. A038003 = Odd Catalan numbers. Cf. A120274, A120275, A119908, A094389.

with(numtheory): c:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: seq(nops(factorset(c(2^n-1))),n=1..15); # Emeric Deutsch, Oct 24 2007

Table[Length[FactorInteger[Binomial[2^(n+1)-2, 2^n-1]/(2^n)]],{n,1,15}]

from sympy import factorint
A119861_list, c, s = [0], {}, 3
for n in range(2,2**19):
    for p,e in factorint(4*n-2).items():
        if p in c:
            c[p] += e
        else:
            c[p] = e
    for p,e in factorint(n+1).items():
        if c[p] == e:
            del c[p]
        else:
            c[p] -= e
    if n == s:
        A119861_list.append(len(c))
        s = 2*s+1 # Chai Wah Wu, Feb 12 2015

a(n) = Length[ FactorInteger[ Binomial[ 2^(n+1)-2, 2^n-1] / (2^n) ]].

a(16)-a(18) from Robert G. Wilson v, May 15 2007

a(19)-a(26) from Chai Wah Wu, Feb 12 2015

A116943 Number of 4s digits plus non-final 3s digits 3 base 5 expansion of 2^n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 3, 2, 2, 2, 1, 1, 3, 4, 5, 3, 3, 1, 4, 4, 7, 2, 7, 7, 4, 6, 9, 9, 6, 5, 5, 7, 4, 9, 4, 7, 7, 7, 10, 8, 6, 8, 6, 9, 8, 9, 8, 10, 11, 11, 8, 13, 5, 11, 15, 13, 10, 10, 8, 12, 9, 14, 11, 8, 11, 12, 10, 13, 13, 13, 10, 10, 12, 6, 10, 15, 8, 17, 17, 16, 16, 12, 16, 15, 13

Offset: 0

Jonathan Vos Post, Mar 23 2006

In his comment on A038003 Frank Adams-Watters conjectures "that 2^n contains such a base 5 digit for n>=9. This is almost certainly true." That is equivalent to a(n) > 0 for n>=9, which is also equivalent to A094389(n) = 5 where A094389 is last decimal digit of the odd Catalan number A038003(n).

			a(7) = 0 because 2^7 (modulo 5) = 1003, which contains 0 digits 4 plus 0 non-final digits 3 (it has a digit 3, but that digit is finial, meaning rightmost).
a(10) = 3 because 2^10 mod 5 = 13044, which contains 2 digits 4 plus 1 non-final digits 3, so 2 + 1 = 3.
a(60) = 10 because 2^60 mod 5 = 34132411211412413323100401, which contains 5 digits 4 plus 5 non-final digits 3, so 5 + 5 = 10.

Cf. A038003, A094389.

f[n_] := Block[{id = IntegerDigits[2^n, 5]}, Count[id, 4] + Count[Most@id, 3]]; Table[ f[n], {n, 0, 88}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Apr 01 2006

A152753 Last digit of even Catalan number A152670(n).

Original entry on oeis.org

2, 4, 2, 2, 0, 2, 6, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 2, 4, 0, 8, 4, 8, 0, 4, 2, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 6, 2, 0, 4, 2, 2, 4, 0, 2, 6, 6, 2, 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: 1

Omar E. Pol, Dec 12 2008

Even values of A152669.

Cf. A000108, A038003, A094389, A152669, A152670.

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A038003 Odd Catalan numbers: a(n) = A000108(2^n-1).

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A119861 Number of distinct prime factors of the odd Catalan numbers A038003(n).

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A116943 Number of 4s digits plus non-final 3s digits 3 base 5 expansion of 2^n.

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A152753 Last digit of even Catalan number A152670(n).

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