cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

Original entry on oeis.org

1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

For denominators see A120777.
From the expansion of 0 = sqrt(1-1) = 1 - (1/2)*Sum_{k>=0} C(k)/4^k one has r:=lim_{n->infinity} r(n) = 2, with the partial sums r(n) defined below.
The series a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 16 2017
If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n-1)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-1)/A000165(2n). See examples. - Anthony Hernandez, Feb 05 2020
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 0..n-1} Catalan(k)/4^k = (1/4^n)*(2*n)*binomial(2*n,n) *( 1 - 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) - 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - ... ). Cf. A082687 and A101028.
This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)

Examples

			Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
From _Anthony Hernandez_, Feb 05 2020: (Start)
For n = 4. The 4th even number is 8, and 8!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
For n = 7. The 7th even number is 14, and 14!!/(14-1)!! = 2048/429, so a(7-1) = a(6) = 2048 - 429 = 1619. (End)

Links

Crossrefs

Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.

Programs

  • Magma
    [Numerator(2*(1-Binomial(2*n+2,n+1)/4^(n+1))): n in [0..25]]; // Vincenzo Librandi, Feb 17 2017
  • Maple
    a := n -> 2^(2*(n+1) - add(i, i=convert(n+1, base, 2)))* (1-((n+1/2)!)/(sqrt(Pi)*(n+1)!)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 21 2017
  • Mathematica
    f[n_] := f[n] = Numerator[(4/Pi) (n + 1) Integrate[x^n*ArcSin[Sqrt[x]], {x, 0, 1}]]; Array[f, 23, 0] (* Robert G. Wilson v, Jan 03 2011 *)
a[n_] := 2^(2(n+1) - DigitCount[n+1,2,1])(1 - ((n+1/2)!)/(Sqrt[Pi](n+1)!));
Table[a[n], {n, 0, 23}] (* Peter Luschny, Dec 21 2017 *)

Formula

a(n) = numerator(r(n)), with the rationals r(n):=Sum_{k = 0..n} C(k)/4^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - Groux Roland, Jan 03 2011
r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - Groux Roland, Jan 04 2011
a(n) = A141244(2n+2) = A141244(2n+3) (conjectural). - Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014
From Peter Luschny, Dec 21 2017: (Start)
a(n) = numerator(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
a(n) = 2^(2*(n+1) - HammingWeight(n+1))*(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)

A086116 Numerator of mean deviation of a symmetrical binomial distribution on n elements.

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 35, 35, 315, 315, 693, 693, 3003, 3003, 6435, 6435, 109395, 109395, 230945, 230945, 969969, 969969, 2028117, 2028117, 16900975, 16900975, 35102025, 35102025, 145422675, 145422675, 300540195, 300540195, 9917826435
Offset: 1

Views

Author

Eric W. Weisstein, Jul 10 2003

Keywords

Comments

From Mohamed Sabba, Apr 24 2023: (Start)
The numerators of this sequence and denominators (A086117) appear in NMR and spin physics as half of the symmetry-constrained upper bound on polarization transfer in AXn spin-1/2 systems. For example:
(a) in AX3 and AX4 spin systems, the maximum achievable transfer from Xz to Az is 3/2 = (2*3/4).
(b) in AX5 and AX6 spin systems, the maximum achievable transfer from Xz to Az is 15/8 = (2*15/16).
Note that this is different from the related adiabatic polarization transfer bounds, given by A141244.
(End)

Links

Crossrefs

Cf. A001803, A086117, A141244.

Programs

  • Mathematica
    Numerator[Table[If[OddQ[n], n!!/2/(n-1)!!, (n-1)!!/2/(n-2)!! ], {n, 50}]]

A141245 Numerators in expansion of (1+x-sqrt(1-x^2))/(x(1-x)).

Original entry on oeis.org

1, 3, 3, 13, 13, 27, 27, 221, 221, 449, 449, 1817, 1817, 3667, 3667, 59101, 59101, 118917, 118917, 478099, 478099, 960397, 960397, 7712569, 7712569, 15477141, 15477141, 62094289, 62094289, 124522883, 124522883, 3994427101
Offset: 0

Views

Author

Paul Barry, Jun 17 2008

Keywords

Comments

The denominators in the expansion of (1+x-sqrt(1-x^2))/(x(1-x)) are 1,2,2,8,8,16,16,....
The sequence 1,3/2,3/2,13/8,13/8,... is the image of 2n+1 under the
Chebyshev related (rational) Riordan array c((x/2)^2),(x/2)c((x/2)^2)) with c(x) the g.f. of A000108.
The Hankel transform of fraction sequence is (-1)^n*(2n+1)/4^comb(n+1,2).

Crossrefs

Cf. A141244.

A362534 Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems.

Original entry on oeis.org

1, 1, 6, 6, 15, 15, 140, 140, 315, 315, 1386, 1386, 3003, 3003, 51480, 51480, 109395, 109395, 92378, 92378, 969969, 969969, 2704156, 2704156, 16900975, 16900975, 70204050, 70204050, 145422675, 145422675, 4808643120, 4808643120, 9917826435, 9917826435, 40838108850, 40838108850
Offset: 1

Views

Author

Mohamed Sabba, Apr 24 2023

Keywords

Comments

In spin physics and NMR, these numbers appear as the numerators of the ratio of different classes of upper bounds on the transfer of z-magnetization in AXn spin systems from a group of spin-1/2 nuclei Xn to a single spin-1/2 nucleus A.
The symmetry-constrained upper bounds are given by the function f(n):
(1) for even n, f(n) = (2^(1-n))*n*binomial(n-1, n/2)
(2) for odd n, f(n) = (2^(1-n))*n*binomial(n-1, (n-1)/2)
The adiabatic bounds are given by the function g(n):
(3) for even n, g(n) = 2*(1-(2^(-n))*binomial(n, n/2))
(4) for odd n, g(n) = 2*(1-(2^(-n))*binomial(n, (n-1)/2))
Where we have the relation:
(5) g(n) = 2*(1 - f(n+1)/(n+1))
The sequence a(n) is defined as the numerator of f(n)/g(n):
(6) a(n) = numerator(f(n)/g(n))
(7) for even n, f(n)/g(n) = (n/2)/(2^(n)*binomial(n, n/2)^(-1) - 1)
(8) for odd n, f(n)/g(n) = ((n+1)/2)/(2^(n)*binomial(n, (n+1)/2)^(-1) - 1)
The first few values of the upper symmetry-constrained bounds f(n) are {1, 1, 3/2, 3/2, 15/8, 15/8, 35/16, 35/16, 315/128, 315/128, ...} which appears to be related to A086116 and A001803.
The first few values of the upper adiabatic bounds g(n) are {1, 1, 5/4, 5/4, 11/8, 11/8, 93/64, 93/64, 193/128, 193/128, ...} which appears to be related to A141244 and A120778.
The first few values of f(n)/g(n) are {1, 1, 6/5, 6/5, 15/11, 15/11, 140/93, 140/93, 315/193, 315/193, ...}
Conjecture: the numerator of g(n) is the denominator of f(n)/g(n).

Links

Crossrefs

Cf. A001803, A086116, A120778, A141244 (denominators but shifted).

Programs

  • Mathematica
    Table[Numerator[Ceiling[n/2]  (2^n Binomial[n, Ceiling[n/2]]^-1 - 1 )^-1], {n, 1, 20}]
  • PARI
    a(n) = numerator(ceil(n/2)/(2^(n)*binomial(n,ceil(n/2))^(-1) - 1)); \\ Michel Marcus, Apr 25 2023

Formula

a(n) = numerator(ceiling(n/2)/(2^(n)*binomial(n,ceiling(n/2))^(-1) - 1)).
Showing 1-4 of 4 results.

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