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A173384 a(n) = A001803(n) - A046161(n). 3
0, 1, 7, 19, 187, 437, 1979, 4387, 76627, 165409, 707825, 1503829, 12706671, 26713417, 111868243, 233431331, 7770342787, 16124087129, 66765132341, 137948422657, 1138049013461, 2343380261227, 9636533415373, 19787656251221 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n-1)!!/(2n-2)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-2)/A000165(2n-1). See examples. - Anthony Hernandez, Feb 05 2020

It can be seen that this is true, e.g., using A001803(n) = (2n+1)!/(n!^2*2^A000120(n)) and A046161(n) = 4^n/2^A000120(n). - M. F. Hasler, Feb 07 2020

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Let r(n) = (-2)^n*Sum_{j=0..n-1} binomial(n,j)*Bernoulli(j+n+1, 1/2)/(j+n+1) then a(n) = numerator(r(n)). - Peter Luschny, Jun 20 2017

EXAMPLE

From Anthony Hernandez, Feb 05 2020: (Start)

Consider n = 4. The 4th odd number is 7, and 7!!/(7-1)!! = 35/16, and a(4-1) = a(3) = 35 - 16 = 19.

Consider n = 7. The 7th odd number is 13, and 13!!/(13-1)!! = 3003/1024, and a(7-1) = a(6) = 3003 - 1024 = 1979. (End)

MAPLE

A046161 := proc(n) binomial(2*n, n)/4^n ; denom(%) ; end proc:

A173384 := proc(n) A001803(n)-A046161(n) ; end proc: # R. J. Mathar, Jul 06 2011

MATHEMATICA

Table[Numerator[(2*n+1)*Binomial[2*n, n]/(4^n)] - Denominator[Binomial[2*n, n]/(4^n)], {n, 0, 30}] (* G. C. Greubel, Dec 09 2018 *)

PROG

(PARI) for(n=0, 30, print1(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/4^n), ", ")) \\ G. C. Greubel, Dec 09 2018

(MAGMA) [Numerator((2*n+1)*Binomial(2*n, n)/(4^n)) - Denominator(Binomial(2*n, n)/(4^n)): n in [0..30]]; // G. C. Greubel, Dec 09 2018

(Sage) [(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/(4^n))) for n in range(30)] # G. C. Greubel, Dec 09 2018

(GAP) List([0..30], n-> (NumeratorRat((2*n+1)*Binomial(2*n, n)/(4^n)) - DenominatorRat(Binomial(2*n, n)/(4^n)))); # G. C. Greubel, Dec 09 2018

CROSSREFS

Cf. A005430.

Sequence in context: A096321 A201806 A128338 * A107195 A201479 A228150

Adjacent sequences:  A173381 A173382 A173383 * A173385 A173386 A173387

KEYWORD

nonn

AUTHOR

Paul Curtz, Feb 17 2010

STATUS

approved

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Last modified September 26 11:29 EDT 2020. Contains 337367 sequences. (Running on oeis4.)