A086116 -id:A086116 - cp's OEIS frontend

A001803 Numerators in expansion of (1 - x)^(-3/2).

1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925

Offset: 0

N. J. A. Sloane

a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). - James R. Buddenhagen, Aug 17 2008
a(n) is the denominator of (2n)!!/(2*n + 1)!! = 2^(2*n)*n!*n!/(2*n + 1)! (see Andersson). - N. J. A. Sloane, Jun 27 2011
a(n) = (2*n + 1)*A001790(n). A046161(n)/a(n) = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (i.e., 1 - 1/3 + 1/5 - 1/7 + ... ). See A173384 and A173396. - Paul Curtz, Feb 21 2010
a(n) is the denominator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding numerator is A101926(n). - Mohammed Yaseen, Jul 25 2023

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.

T. D. Noe, Table of n, a(n) for n = 0..200
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mats Erik Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Eric Weisstein's World of Mathematics, Heads-Minus-Tails Distribution, Random Walk--1-Dimensional, Circle Line Picking.

The denominator is given in A046161.
Largest odd divisors of A001800, A002011, A002457, A005430, A033876, A086228.
Bisection of A004731, A004735, A086116.
Second column of triangle A100258.
Cf. A161199, A161201, A173384, A173396.
Cf. A002596 (numerators in expansion of (1-x)^(1/2)).
Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)).
A163590 is the odd part of the swinging factorial, A001790 at even indices. - Peter Luschny, Aug 01 2009

A001803(n) = sum(<<(A001790(k), A005187(n) - A005187(k)) for k in 0:n) # Peter Luschny, Oct 03 2019

A001803:= func< n | Numerator(Binomial(n+2,2)*Catalan(n+1)/4^n) >;
[A001803(n): n in [0..30]]; // G. C. Greubel, Apr 27 2025

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
a := n -> swing(2*n+1)/sigma(2*n+1); # Peter Luschny, Aug 01 2009
A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011
a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # Peter Luschny, Dec 06 2024

Numerator/@CoefficientList[Series[(1-x)^(-3/2),{x,0,25}],x]  (* Harvey P. Dale, Feb 19 2011 *)
Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)

a(n) = numerator((2*n+1)*binomial(2*n,n)/(4^n)); \\ Altug Alkan, Sep 06 2018

def A001803(n): return numerator((n+1)*binomial(2*n+2,n+1)/2^(2*n+1))
print([A001803(n) for n in range(31)]) # G. C. Greubel, Apr 27 2025

a(n) = (2*n + 1)! /(n!^2*2^A000120(n)) = (n + 1)*binomial(2*n+2,n+1)/2^(A000120(n)+1). - Ralf Stephan, Mar 10 2004
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = numerator( (2*n+1)*binomial(2*n,n)/(4^n) ).
(1 - x)^(-3/2) = Sum_{n>=0} ((2*n+1)*binomial(2*n,n)/4^n)*x^n. (End)
Truncations of rational expressions like those given by the numerator or denominator operators are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - Peter Luschny, Aug 01 2009
a(n) = denominator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 06 2024

A086117 Denominator of mean deviation of a symmetrical binomial distribution on n elements.

Original entry on oeis.org

2, 2, 4, 4, 16, 16, 32, 32, 256, 256, 512, 512, 2048, 2048, 4096, 4096, 65536, 65536, 131072, 131072, 524288, 524288, 1048576, 1048576, 8388608, 8388608, 16777216, 16777216, 67108864, 67108864, 134217728, 134217728, 4294967296, 4294967296

Offset: 1

Eric W. Weisstein, Jul 10 2003

Also denominator of x(n)=Sum(x(k)*x(n-k-1):0<=kA098597(n)/a(n+2). - Reinhard Zumkeller, Feb 06 2008

Eric Weisstein's World of Mathematics, Binomial Distribution
Index to divisibility sequences

Cf. A086116, A113474.

Twice A101926.

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

a(1)=2, a(n) = a(n-1)*A006519(n). - Jon Perry, Mar 31 2004

a(n) = 2^A113474(n-1). - Alan Michael Gómez Calderón, Apr 03 2025

A366568 P(n,n+2k-2) are the numerators of the probability that a gambler with initial capital n goes bankrupt exactly at round n+2k-2, randomly losing or winning 1 unit of money in each round, written as lower triangle T(n,k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 5, 1, 1, 9, 7, 7, 1, 5, 7, 7, 21, 21, 1, 3, 5, 3, 45, 33, 33, 1, 7, 27, 75, 165, 297, 429, 429, 1, 1, 35, 55, 275, 143, 1001, 715, 715, 1, 9, 11, 77, 429, 1001, 1001, 429, 2431, 2431

Offset: 1

Hugo Pfoertner, Oct 24 2023

			The triangle begins
  1;
  1, 1;
  1, 1, 1;
  1, 3, 5, 5;
  1, 1, 9, 7,  7;
  1, 5, 7, 7, 21, 21;
  1, 3, 5, 3, 45, 33, 33;
The numbering is as follows, with
 [n, rounds, P] = [initial capital, coin flips, probability of ruin]:
  [1, 1,  1/2]
  [2, 2,  1/4] [1, 3,   1/8]
  [3, 3,  1/8] [2, 4,   1/8] [1, 5,  1/16]
  [4, 4, 1/16] [3, 5,  3/32] [2, 6,  5/64] [1, 7, 5/128]
  [5, 5, 1/32] [4, 6,  1/16] [3, 7, 9/128] [2, 8, 7/128] [1,  9,  7/256]
  [6, 6, 1/64] [5, 7, 5/128] [4, 8, 7/128] [3, 9, 7/128] [2, 10, 21/512]
                                                           [1, 11, 21/1024]
With initial capital odd, ruin can only occur at odd numbered rounds, with even initial capital only at even numbered rounds.

Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
Eric Weisstein's World of Mathematics, Random Walk--1-Dimensional.
Wikipedia, One-dimensional random walk

A366569 are the corresponding denominators.

Cf. A086116, A141387.

P(m,mue) = (mue/m) * mue! / (((mue - m)/2)! * ((mue + m)/2)!) * (1/2)^mue, with round count mue and initial capital m (Bachelier, 1912, page 103).

P(m,mue) = 0 for mue < m and for mue + m odd.

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

Mohamed Sabba, Apr 24 2023

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).

G. C. Chingas, A. N. Garroway, W. B. Moniz, and R. D. Bertrand, Adiabatic J cross-polarization in liquids for signal enhancement in NMR, Journal of Chemical Physics, 102:8 (1980), 2526-2528 (page 1, equation 2 gives an expression for the adiabatic bounds).
Malcolm H. Levitt, Symmetry constraints on spin dynamics: Application to hyperpolarized NMR, Journal of Chemical Physics, 102:8 (2016).
Ole W. Sørensen, A universal bound on spin dynamics, Journal of Magnetic Resonance, 262 (1990) (appendix gives proof of the expression for symmetry constrained bounds).

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

Table[Numerator[Ceiling[n/2]  (2^n Binomial[n, Ceiling[n/2]]^-1 - 1 )^-1], {n, 1, 20}]

a(n) = numerator(ceil(n/2)/(2^(n)*binomial(n,ceil(n/2))^(-1) - 1)); \\ Michel Marcus, Apr 25 2023

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

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A001803 Numerators in expansion of (1 - x)^(-3/2).

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A086117 Denominator of mean deviation of a symmetrical binomial distribution on n elements.

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A366568 P(n,n+2k-2) are the numerators of the probability that a gambler with initial capital n goes bankrupt exactly at round n+2k-2, randomly losing or winning 1 unit of money in each round, written as lower triangle T(n,k), n>=1, 1<=k<=n.

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

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