A101926 -id:A101926 - cp's OEIS frontend

A001790 Numerators in expansion of 1/sqrt(1-x).

1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719

Offset: 0

N. J. A. Sloane

Also numerator of e(n-1,n-1) (see Maple line).
Leading coefficient of normalized Legendre polynomial.
Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005
This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007
Truncations of rational expressions like those given by the numerator operator 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)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009
It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - James R. Buddenhagen, Jan 20 2010
The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012
a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013
Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016
From Ralf Steiner, Apr 07 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A060818(k) = -i.
Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3).
a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2).
Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End)
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/A344402. - Peter Luschny, May 20 2021
Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022
a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023
a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023

			1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
binomial(2*n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
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 6:14:6 at page 51.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.

Robert G. Wilson v, Table of n, a(n) for n = 0..1666 (first 201 terms from T. D. Noe)
Horst Alzer and Bent Fuglede, Normalized binomial mid-coefficients and power means, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294.
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993 (see V_n with N=1).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table [Annotated scanned copy].
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
S. Łukaszyk and W. Bieniawski, Assembly Theory of Binary Messages, Mathematics, 12(10) (2024), 1600.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
Eric Weisstein's World of Mathematics, Binomial Series.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Lorentz Factor.

Cf. A000142, A000984, A001147, A001316, A001800, A001801, A001803.
Cf. A005187, A007814, A008316, A046161, A056040, A086117, A094638.
Cf. A101926, A123854, A273194, A344402.
Cf. A060818 (denominator of binomial(2*n,n)/2^n), A061549 (denominators).
Cf. A123854 (denominators).
Cf. A161198 (triangle of coefficients for (1-x)^((-1-2*n)/2)).
Cf. A163590 (odd part of the swinging factorial).
Cf. A001405.
First column and diagonal 1 of triangle A100258.
Bisection of A036069.
Bisections give A061548 and A063079.
Inverse Moebius transform of A180403/A046161.
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), this sequence (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;
[A001790(n): n in [0..40]]; // G. C. Greubel, Sep 23 2024

e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end;
# From Peter Luschny, Aug 01 2009: (Start)
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)/sigma(2*n); # (End)
A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # R. J. Mathar, Jan 18 2013

Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}]   (* John M. Campbell, Jul 04 2011 *)
Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]),{n,0,20}]] (* Ralf Steiner, Apr 07 2017 *)
Numerator[Table[Binomial[2n,n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *)
A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* John Lawrence, Jul 17 2020 *)

{a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};

a(n)=binomial(2*n,n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021

# uses[A000120]
@CachedFunction
def swing(n):
    if n == 0: return 1
    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
A001790 = lambda n: swing(2*n)/2^A000120(2*n)
[A001790(n) for n in (0..25)]  # Peter Luschny, Nov 19 2012

a(n) = numerator( binomial(2*n,n)/4^n ) (cf. A046161).
a(n) = A000984(n)/A001316(n) where A001316(n) is the highest power of 2 dividing C(2*n, n) = A000984(n). - Benoit Cloitre, Jan 27 2002
a(n) = denominator of (2^n/binomial(2*n,n)). - Artur Jasinski, Nov 26 2011
a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0.
Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013
From Ralf Steiner, Apr 08 2017: (Start)
a(n)/A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A061549(k) = 2/sqrt(3).
Sum_{k>=0} (-1)^k*a(k)/A061549(k) = 2/sqrt(5).
Sum_{k>=0} (-1)^(k+1)*a(k)/A061549(k) = -2/sqrt(5).
a(n)/A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)).
Sum_{k>=0} a(k)/A123854(k) = sqrt(2).
Sum_{k>=0} (-1)^k*a(k)/A123854(k) = sqrt(2/3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A123854(k) = -sqrt(2/3). (End)
a(n) = 2^A007814(n)*(2*n-1)*a(n-1)/n. - John Lawrence, Jul 17 2020
Sum_{k>=0} A086117(k+3)/a(k+2) = Pi. - Antonio Graciá Llorente, Aug 31 2024
a(n) = A001803(n)/(2*n+1). - G. C. Greubel, Sep 23 2024

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

Original entry on oeis.org

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

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

Original entry on oeis.org

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000

Offset: 0

N. J. A. Sloane

Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.

T. D. Noe, Table of n, a(n) for n = 0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.

Cf. A000165, A001818, A079484, A197036, A334380.

J1: A002474, J2: A002506, J3: A014401.

[4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019

Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)

a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019

(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022

A101925 a(n) = A005187(n) + 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

Offset: 0

Ralf Stephan, Dec 28 2004

nonn

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Bisection of A089279. First differences are in A001511.
Cf. A000120, A005187, A046699.
Cf. A032184, A052278, A060055, A066318, A088229, A101926.

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PARI
```
a(n)=1+sum(k=1, n, valuation(k,2)+1)
```

a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))

a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022
(Python 3.10+)
def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864, 300540195, 2147483648, 9917826435, 4294967296, 20419054425

Offset: 0

N. J. A. Sloane

Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).
Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of x(n+2) equals the denominator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010, as corrected in Cooper (2015).
Numerator of (n-1)/( (n-2)/( .../1)), with an empty fraction taken to be 1. - Flávio V. Fernandes, Jan 31 2025

			1, 1, (1/2)*Pi, 2, (3/4)*Pi, 8/3, (15/16)*Pi, 16/5, (35/32)*Pi, 128/35, (315/256)*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), 1/2*Pi^(1/2), 2/Pi^(1/2), 3/4*Pi^(1/2), 8/3/Pi^(1/2), 15/16*Pi^(1/2), 16/5/Pi^(1/2), ...

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

T. D. Noe, Table of n, a(n) for n=0..302
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Svante Janson, On the traveling fly problem, Graph Theory Notes of New York Vol. XXXI, 17, 1996.

Cf. A001803, A004730, A006882 (double factorials), A036069.

Cf. A036039, A046161, A001790, A001803, A101926.

import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
   ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1,k)/4^k else k := (n-1)/2; 4^k/binomial(2*k,k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));
#
[1, seq(denom(doublefactorial(n-2)/doublefactorial(n-1)), n = 1..36)]; # Peter Luschny, Feb 09 2025

Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* Jean-François Alcover, Jul 16 2012 *)
Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2,40]!!,2,1]] (* Harvey P. Dale, Nov 27 2014 *)
Join[{1},Table[Numerator[(n/2-1/2)!/((n/2-1)!Sqrt[Pi])], {n,1,31}]] (* Peter Luschny, Feb 08 2025 *)

f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
a(n) = if (n==0, 1, denominator(f(n-2)/f(n-1))); \\ Michel Marcus, Feb 08 2025

from sympy import gcd, factorial2
def A004731(n):
    if n <= 1:
        return 1
    a, b = factorial2(n-2), factorial2(n-1)
    return b//gcd(a,b) # Chai Wah Wu, Apr 03 2021

Name corrected by Michel Marcus, Feb 08 2025

A139566 a(n) is the sum of squares of digits of a(n-1); a(1)=15.

Original entry on oeis.org

15, 26, 40, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37

Offset: 1

Robert Gornall (rob(AT)khobbits.net), Jun 11 2008

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A101926, A087965, A074411, A110998, A051861, A048222.

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4), A000221 (starting with 5), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

a = {15}; Do[AppendTo[a, Plus @@ (IntegerDigits[a[[ -1]]]^2)], {70}]; a (* Stefan Steinerberger, Jun 14 2008 *)
NestList[Total[IntegerDigits[#]^2] &, 15, 70] (* or *) PadRight[ {15,26,40},70,{42,20,4,16,37,58,89,145}](* Harvey P. Dale, Jan 28 2013 *)

/* to check the given terms */
a=[/* paste the terms here */]; a==vector(#a,n,k=if(n>1,A003132(k),15))
/* to check the following code, use: a==vector(99,n,A139566(n)) */
A139566(n)=[15,26,40,16,37,58,89,145,42,20,4][if(n>11,(n-4)%8+4,n)] \\ (End)

Vec(x*(36*x^10+6*x^9-27*x^8-145*x^7-89*x^6-58*x^5-37*x^4-16*x^3 -40*x^2-26*x-15)/((x-1)*(x+1)*(x^2+1)*(x^4+1)) + O(x^70)) \\ Colin Barker, Aug 24 2015

Eventually periodic with period 8.
a(n) = A008463(n) for n > 4. - M. F. Hasler, May 24 2009
a(n) = a(n-8) for n > 11. - Colin Barker, Aug 24 2015
G.f.: x*(36*x^10 + 6*x^9 - 27*x^8 - 145*x^7 - 89*x^6 - 58*x^5 - 37*x^4 - 16*x^3 - 40*x^2 - 26*x - 15) / ((x-1)*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Aug 24 2015

More terms from Stefan Steinerberger, Jun 14 2008
Terms checked, using the given PARI code, by M. F. Hasler, May 24 2009
Minor edits and starting value added in name by M. F. Hasler, Apr 27 2018

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

A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).

Original entry on oeis.org

1, 2, 1, 4, 3, 16, 5, 32, 35, 256, 63, 512, 231, 2048, 429, 4096, 6435, 65536, 12155, 131072, 46189, 524288, 88179, 1048576, 676039, 8388608, 1300075, 16777216, 5014575, 67108864, 9694845, 134217728, 300540195

Offset: 0

N. J. A. Sloane

Also rational part of denominator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A004731).

			1, 1, 1/2*Pi, 2, 3/4*Pi, 8/3, 15/16*Pi, 16/5, 35/32*Pi, 128/35, 315/256*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), (1/2)*Pi^(1/2), 2/Pi^(1/2), (3/4)*Pi^(1/2), (8/3)/Pi^(1/2), (15/16)*Pi^(1/2), (16/5)/Pi^(1/2), ...

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

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

Cf. A004731.
Bisections are A001790 and A101926.
Cf. A004731, A046161, A001790, A001803, A101926.

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1,k)/4^k else k := (n-1)/2; 4^k/binomial(2*k,k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));

Table[ Denominator[ Gamma[n/2+1]/Gamma[n/2+1/2]*Sqrt[Pi]^(1 - 2 Mod[n, 2])], {n, 0, 32}] (* Jean-François Alcover, Jul 16 2012 *)

A144816 Denominators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2*k+1) in polynomial t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

Original entry on oeis.org

1, 2, 2, 8, 4, 8, 16, 16, 16, 16, 128, 32, 64, 32, 128, 256, 256, 128, 128, 256, 256, 1024, 512, 1024, 256, 1024, 512, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 32768, 4096, 8192, 4096, 16384, 4096, 8192, 4096, 32768, 65536, 65536, 16384, 16384, 32768, 32768, 16384, 16384, 65536, 65536

Offset: 0

Alois P. Heinz, Sep 21 2008

			Triangle begins:
    1;
    2,  2;
    8,  4,  8;
   16, 16, 16, 16;
  128, 32, 64, 32, 128;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

See A144815 for more information on T(n,k).
Main diagonal and column k=0 gives A046161.
Column k=1 gives A101926(n-1) = 2^A101925(n-1) = 2^(A005187(n-1)+1).
Cf. A077070.

# Function T(n,k) defined in A144815.
seq(seq(denom(T(n,k)), k=0..n), n=0..10);

row[n_] := Module[{f, a, eq}, f = Function[x, Sum[a[2*k+1]*x^(2*k+1), {k, 0, n}]]; eq = Table[Derivative[k][f][1] == If[k == 0, 1, 0], {k, 0, n}]; Table[a[2*k+1], {k, 0, n}] /. Solve[eq] // First]; Table[row[n] // Denominator, {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 03 2014 *)

A173396 a(n) = A046161(n) + A001803(n).

Original entry on oeis.org

2, 5, 23, 51, 443, 949, 4027, 8483, 142163, 296481, 1232113, 2552405, 21095279, 43490633, 178977107, 367649059, 12065310083, 24714021721, 101124870709, 206667899393, 1687804827349, 3442891889003, 14034579926477, 28583749273429

Offset: 0

Paul Curtz, Feb 17 2010

nonn

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

Cf. A173384, A173294, A173296.

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

[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

A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc:
A046161 := proc(n) binomial(2*n,n)/4^n ; denom(%) ; end proc:
A173386 := proc(n) A001803(n)+A046161(n) ; end proc: # R. J. Mathar, Jul 06 2011

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 *)

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

[(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

a(n) = A101926(n) + A173384(n).

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A001790 Numerators in expansion of 1/sqrt(1-x).

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

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A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

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A101925 a(n) = A005187(n) + 1.

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A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

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A139566 a(n) is the sum of squares of digits of a(n-1); a(1)=15.

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