A173384 - OEIS

%I #56 Apr 13 2024 16:05:30

%S 0,1,7,19,187,437,1979,4387,76627,165409,707825,1503829,12706671,

%T 26713417,111868243,233431331,7770342787,16124087129,66765132341,

%U 137948422657,1138049013461,2343380261227,9636533415373,19787656251221

%N a(n) = 2^(2*n - HammingWeight(n)) * [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

%C 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

%C 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

%C Numerators in the expansion of (1-(1-x)^(1/2))/(1-x)^(3/2). Denominators are A046161. Compare to A001790. - _Thomas Curtright_, Feb 09 2024

%H G. C. Greubel, <a href="/A173384/b173384.txt">Table of n, a(n) for n = 0..1000</a>

%H Thomas Curtright and Gaurav Verma, <a href="https://doi.org/10.48550/arXiv.2404.07745">Scattering Shadows</a>, arXiv:2404.07745 [physics.class-ph], 2024, p. 9.

%F a(n) = A001803(n) - A046161(n).  (Previous name.)

%F 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

%e From _Anthony Hernandez_, Feb 05 2020: (Start)

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

%e 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)

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

%p A173384 := proc(n) A001803(n)-A046161(n) ; end proc: # _R. J. Mathar_, Jul 06 2011

%t 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 *)

%t A173384[n_] := 2^(2*n - DigitCount[n, 2, 1]) Coefficient[Series[(x - 1)^(-1) + (1 - x)^(-3/2), {x, 0, n}], x, n]

%t Table[A173384[n], {n, 0, 23}]  (* _Peter Luschny_, Feb 17 2024 *)

%o (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

%o (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

%o (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

%o (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

%Y Cf. A001790, A005430, A046161.

%K nonn

%O 0,3

%A _Paul Curtz_, Feb 17 2010

%E New name using an expansion of _Thomas Curtright_ by _Peter Luschny_, Feb 17 2024