A173384 a(n) = 2^(2*n - HammingWeight(n)) * [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

0, 1, 7, 19, 187, 437, 1979, 4387, 76627, 165409, 707825, 1503829, 12706671, 26713417, 111868243, 233431331, 7770342787, 16124087129, 66765132341, 137948422657, 1138049013461, 2343380261227, 9636533415373, 19787656251221

Offset: 0

Author

Paul Curtz, Feb 17 2010

Keywords

• nonn

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

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

Examples

			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)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Thomas Curtright and Gaurav Verma, Scattering Shadows, arXiv:2404.07745 [physics.class-ph], 2024, p. 9.

Crossrefs

Cf. A001790, A005430, A046161.

Programs

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

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

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 *)
A173384[n_] := 2^(2*n - DigitCount[n, 2, 1]) Coefficient[Series[(x - 1)^(-1) + (1 - x)^(-3/2), {x, 0, n}], x, n]
Table[A173384[n], {n, 0, 23}]  (* Peter Luschny, Feb 17 2024 *)

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

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

Formula

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

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

Extensions

New name using an expansion of Thomas Curtright by Peter Luschny, Feb 17 2024