A046161 -id:A046161 - 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

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

Original entry on oeis.org

1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35

Offset: 1

Johannes W. Meijer, Dec 24 2012

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

Cf. A000027 (the natural numbers), A000120 (1's-counting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).

-- Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n-1)
a220466_list = 1 : concat
   (transpose [zipWith (-) (map (* 4) a220466_list) a006519_list, [2..]])
-- Reinhard Zumkeller, Aug 31 2014

# First Maple program
a := n -> 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n1-1)+2^(p-1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od:  seq(a(n), n=1..nmax);

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n - 1) + 2^(p - 1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))
a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))
Recurrence: a(2n) = 4a(n) - 2^A007814(n), a(2n+1) = n+1. - Ralf Stephan, Dec 17 2013

A056982 a(n) = 4^A005187(n). The denominators of the Landau constants.

Original entry on oeis.org

1, 4, 64, 256, 16384, 65536, 1048576, 4194304, 1073741824, 4294967296, 68719476736, 274877906944, 17592186044416, 70368744177664, 1125899906842624, 4503599627370496, 4611686018427387904, 18446744073709551616, 295147905179352825856, 1180591620717411303424

Offset: 0

Eric W. Weisstein

Also equal to A046161(n)^2.
Let W(n) = Product_{k=1..n} (1- 1/(4*k^2)), the partial Wallis product with lim n -> infinity W(n) = 2/Pi; a(n) = denominator(W(n)). The numerators are in A069955.
Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n >= 1} 4*n^2/(4*n^2-1). Numerators are in A069955.
Denominator of h^(2n) in the Kummer-Gauss series for the perimeter of an ellipse.
Denominators of coefficients in hypergeometric([1/2,-1/2],[1],x). The numerators are given in A038535. hypergeom([1/2,-1/2],[1],e^2) = L/(2*Pi*a) with the perimeter L of an ellipse with major axis a and numerical eccentricity e (Maclaurin 1742). - Wolfdieter Lang, Nov 08 2010
Also denominators of coefficients in hypergeometric([1/2,1/2],[1],x). The numerators are given in A038534. - Wolfdieter Lang, May 29 2016
Also denominators of A277233. - Wolfdieter Lang, Nov 16 2016
A277233(n)/a(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 84. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
O. J. Farrell and B. Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Charles R Greathouse IV, Table of n, a(n) for n = 0..500
B. Gourevitch, L'univers de Pi
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the Hathi Trust Digital Library.]
Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.
G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.
Eric Weisstein's World of Mathematics, Gauss-Kummer Series
Eric Weisstein's World of Mathematics, Ellipse
Index to divisibility sequences

Apart from offset, identical to A110258.
Cf. A005187, A046161, A056981, A069955.
Equals (1/2)*A038533(n), A038534, A277233.
Cf. A001790/A046161.

A056982 := n -> denom(binomial(1/2, n))^2:
seq(A056982(n), n=0..19); # Peter Luschny, Apr 08 2016
# Alternatively:
G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser,x,n), n=0..19)]: denom(%); # Peter Luschny, Sep 28 2019

Table[Power[4, 2 n - DigitCount[2 n, 2, 1]], {n, 0, 19}] (* Michael De Vlieger, May 30 2016, after Harvey P. Dale at A005187 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Denominator (* Peter Luschny, Sep 28 2019 *)

a(n)=my(s=n); while(n>>=1, s+=n); 4^s \\ Charles R Greathouse IV, Apr 07 2012

a(n) = (denominator(binomial(1/2, n)))^2. - Peter Luschny, Sep 27 2019

Edited by N. J. A. Sloane, Feb 18 2004, Jun 05 2007

A060818 a(n) = 2^(n - HammingWeight(n)) = 2^(n - BitCount(n)) = 2^(n - A000120(n)).

Original entry on oeis.org

1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, 256, 1024, 1024, 2048, 2048, 32768, 32768, 65536, 65536, 262144, 262144, 524288, 524288, 4194304, 4194304, 8388608, 8388608, 33554432, 33554432, 67108864, 67108864, 2147483648, 2147483648, 4294967296

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001

a(n) is the size of the Sylow 2-subgroup of the symmetric group S_n.
Also largest power of 2 which is a factor of n! and (apart from a(3)) the largest perfect power which is a factor of n!.
Denominator of e(n,n) (see Maple line).
Denominator of the coefficient of x^n in n-th Legendre polynomial; numerators are in A001790. - Benoit Cloitre, Nov 29 2002

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 16*x^7 + 128*x^8 + ...
e(n,n) sequence begins 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ... .

Harry J. Smith, Table of n, a(n) for n = 0..200
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, 87(2) (2014), 135-143.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49(3) (2002), 311-317.
Eric Weisstein's World of Mathematics, Random Walk 1-Dimensional.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Index to divisibility sequences

Cf. A000120, A005187, A001790, A011371, A334907.
a(n) = A046161([n/2]).
Row sums of triangle A100258.

[1] cat [Denominator(Catalan(n)/2^n): n in [0..50]]; // Vincenzo Librandi, Sep 01 2014
(Python 3.10+)
def A060818(n): return 1<Chai Wah Wu, Jul 11 2022

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;
A060818 := proc(n) option remember; `if`(n=0,1,2^(padic[ordp](n,2))*A060818(n-1)) end: seq(A060818(i), i=0..34); # Peter Luschny, Nov 16 2012
HammingWeight := n -> add(convert(n, base, 2)):
seq(2^(n - HammingWeight(n)), n = 0..34); # Peter Luschny, Mar 23 2024

Table[GCD[w!, 2^w], {w, 100}]
(* Second program, more efficient *)
Array[2^(# - DigitCount[#, 2, 1]) &, 35, 0] (* Michael De Vlieger, Mar 23 2024 *)

{a(n) = denominator( polcoeff( pollegendre(n), n))};

{a(n) = if( n<0, 0, 2^sum(k=1, n, n\2^k))};

{ for (n=0, 200, s=0; d=2; while (n>=d, s+=n\d; d*=2); write("b060818.txt", n, " ", 2^s); ) } \\ Harry J. Smith, Jul 12 2009

def A060818(n):
    A005187 = lambda n: A005187(n//2) + n if n > 0 else 0
    return 2^A005187(n//2)
[A060818(i) for i in (0..34)]  # Peter Luschny, Nov 16 2012

a(n) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...).
a(n) = 2^(A011371(n)).
a(n) = gcd(n!, 2^n). - Labos Elemer, Apr 22 2003
a(n) = denominator(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 factorial (2*n-1)!! = A001147(n), n>=0.
a(n) = Product_{i=1..n} A006519(i). - Tom Edgar, Apr 30 2014
a(n) = (n! XOR floor(n!/2)) XOR (n!-1 XOR floor((n!-1)/2)). - Gary Detlefs, Jun 13 2014
a(n) = denominator(Catalan(n-1)/2^(n-1)) for n>0. - Vincenzo Librandi, Sep 01 2014
a(2*n) = a(2*n+1) = 2^n*a(n). - Robert Israel, Sep 01 2014
a(n) = n!*A063079(n+1)/A334907(n). - Petros Hadjicostas, May 16 2020

Additional comments from Henry Bottomley, May 01 2001

New name from Peter Luschny, Mar 23 2024

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128

Offset: 0

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2, .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(x,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.
Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.
These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.

			From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of matrix M are:
  1, 2,  0,  0, 0, ...
  1, 3,  2,  0, 0, ...
  1, 4,  5,  2, 0, ...
  1, 5,  9,  7, 2, ...
  1, 6, 14, 16, 9, ... (End)
The first few G(p,n) polynomials are:
  G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
  G(p,-2) = 3 - 8*p + 4*p^2
  G(p,-1) = 1 - 2*p
The first few F(p,n) polynomials are:
  F(p,0) = 1
  F(p,1) = 1 + 2*p
  F(p,2) = 3 + 8*p + 4*p^2
  F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
The first few rows of the upper and lower hourglass triangles are:
  [15, -46, 36, -8]
  [3, -8, 4]
  [1, -2]
  [1]
  [1, 2]
  [3, 8, 4]
  [15, 46, 36, 8]

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].
A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).
A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.
A001147 is the first left hand column and equals the row sums.
A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.
A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.
A074599 * A025549 is not always equals the second left hand column.
Cf. A029635. [Gary W. Adamson, Jul 19 2011]

nmax:=7; for n from 0 to nmax do a(n,n):=2^n: a(n,0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n,m) := 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) od: od: seq(seq(a(n,k), k=0..n), n=0..nmax);
nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): A029635 := proc(n,k): binomial(n,k) + binomial(n-1,k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := A029635(i,j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n,m):= B[1,m+1] od: od: seq(seq(a(n,m), m=0..n), n=0..nmax);
A161198 := proc(n,k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:
seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

for(n=0,9, print(Vec(Ser( 2^n*prod( k=1,n, x+(2*k-1)/2 ),,n+1))))  \\ M. F. Hasler, Jul 23 2011

@CachedFunction
def A161198(n,k):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
for n in (0..6): [A161198(n,k) for k in (0..n)]  # Peter Luschny, May 09 2013

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .
a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.
a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]
T(n,k) = 2^k * A028338(n,k). - Philippe Deléham, May 14 2015

A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).

Original entry on oeis.org

1, 1, -1, 1, -5, 7, -21, 33, -429, 715, -2431, 4199, -29393, 52003, -185725, 334305, -9694845, 17678835, -64822395, 119409675, -883631595, 1641030105, -6116566755, 11435320455, -171529806825, 322476036831, -1215486600363, 2295919134019

Offset: 0

N. J. A. Sloane

Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)-th Catalan number.
From Dimitri Papadopoulos, Oct 28 2016: (Start)
The sum of the coefficients of the expansion of sqrt(1+x) is sqrt(2) (easy). Observation: The sum of the squares of the coefficients is 4/Pi.
Observation/conjecture: If a term of this sequence is divisible by a prime p, then that term is in a block of exactly (p^k-3)/2 consecutive terms all of which are divisible by p. Furthermore, if a(n) is the term preceding such a block then a(p*n-(p-1)/2) also precedes a block of (p^(k+1)-3)/2 terms all divisible by p.
E.g., a(4)=-5 is divisible by 5 and is in a block of (5^1 - 3)/2 = 1 consecutive terms that are all divisible by 5. Then a(5*3 - (5-1)/2) = a(13) = 52003 precedes a block of exactly (5^2 - 3)/2 = 11 terms all divisible by 5.
(End)

			sqrt(1+x) = 1 + (1/2)*x - (1/8)*x^2 + (1/16)*x^3 - (5/128)*x^4 + (7/256)*x^5 - (21/1024)*x^6 + (33/2048)*x^7 + ...
Coefficients are 1, 1/2, -1/8, 1/16, -5/128, 7/256, -21/1024, 33/2048, -429/32768, 715/65536, -2431/262144, 4199/524288, -29393/4194304, 52003/8388608, ...

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.281).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 88.
Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 72.
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.

T. D. Noe, Table of n, a(n) for n = 0..200
Tom Copeland, Addendum to Elliptic Lie Triad.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Denominators are A046161.
Cf. A001795.
Equals A000265(A000108(n-1)), n>0.
Absolute values are essentially A098597.
From Johannes W. Meijer, Jun 08 2009: (Start)
Cf. A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)], A001803 [(1-x)^(-3/2)].
Cf. A161198 = triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n. (End)

[(-1)^n*Numerator((1/(1-2*n))*Binomial(2*n,n)/(4^n)): n in [0..30]]; // Vincenzo Librandi, Jan 14 2016

seq(numer(subs(k=1/2,expand(binomial(k,n)))),n=0..50); # James R. Buddenhagen, Aug 16 2014

1+InverseSeries[Series[2^p*y+y^2/2^q, {y, 0, 24}], x] (* p, q positive integers, then a(n)=numerator(y(n)). - Len Smiley, Apr 13 2000 *)
Numerator[CoefficientList[Series[Sqrt[1+x],{x,0,30}],x]] (* Harvey P. Dale, Oct 22 2011 *)
Table[Numerator[Product[(3 - 2 k)/(2 k) , {k, j}]], {j, 0, 30}] (* Dimitri Papadopoulos, Oct 22 2016 *)

x = 'x + O('x^40); apply(x->numerator(x), Vec(sqrt(1+x))) \\ Michel Marcus, Jan 14 2016

a(n+2) = C(n+1)/2^k(n+1), n >= 0; where C(n) = A000108(n), k(n) = A048881(n).
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = (-1)^n*numerator((1/(1-2*n))*binomial(2*n,n)/(4^n)).
(1+x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(-x)^n.
(1-x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(x)^n. (End)
a(n) = numerator(Product_{k=1..n} (3-2*k)/(2*k)). - Dimitri Papadopoulos, Oct 22 2016

Minor correction to definition from Johannes W. Meijer, Jun 05 2009

A171977 a(n) = 2^(k+1) where 2^k is the highest power of 2 dividing n.

Original entry on oeis.org

2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 64, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2

Offset: 1

Paul Curtz, Nov 19 2010

When read as a triangle in which the n-th row has 2^n terms, every row is the last half of the next one. All the terms are powers of 2. First column = 2*A000079.
The original definition was: a(n) = (A000265(2n+1) - 1) / A000265(2n).
a(n) seems to be the denominator of Euler(2*n+1,1) but I have no proof of this.
a(n) is also gcd[C(2n,1), C(2n,3), ..., C(2n,2n-1)]. - Franz Vrabec, Oct 22 2012
a(n) is also the ratio r(2n) = s2(2n)/s1(2n) where s1(2n) is the sum of the odd unitary divisors of 2n and s2(2n) is the sum of the even unitary divisors of 2n. - Michel Lagneau, Dec 19 2013
a(n) or a(n)/2 = A006519(n) is known as the Steinhaus sequence in probability theory, proposed as a sequence of asymptotically fair premiums for the St. Petersburg game. - Peter Kern, Aug 28 2015
After the all-1's sequence this is the next sequence in lexicographical order such that the gap between a(n) and the next occurrence of a(n) is given by a(n). - Scott R. Shannon, Oct 16 2019
First 2^(k-1) - 1 terms are also the areas of the successive rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure of A139250 after 2^k stages, with k >= 2. For example: if k = 5 the areas after 32 stages are [2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2] respectively, the same as the first 15 terms of this sequence. - Omar E. Pol, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383
Sandor Csörgö and Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157--172. MR1321758 (96b:60017). - _Peter Kern_, Aug 28 2015
Roger B. Eggleton, Aviezri S. Fraenkel, and R. Jamie Simpson, Beatty sequences and Langford sequences, Graph theory and combinatorics (Marseille-Luminy, 1990). Discrete Math. 111 (1993), no. 1-3, 165--178. MR1210094 (94a:11018). See Example 2.6. - _N. J. A. Sloane_, Mar 18 2012
Hugo Steinhaus, The so-called Petersburg paradox, Colloq. Math. 2 (1949), 56--58. MR0039937 (12,619e).

Cf. A000079, A000265, A006519, A038712, A139250.

a := proc(n) local k: k:=1: while frac(n/2^k) = 0 do k := k+1 end do: k := k-1: a(n) := 2^(k+1) end: seq(a(n), n=1..63); # Johannes W. Meijer, Nov 04 2012
seq(2^(1 + padic[ordp](n, 2)), n = 1..63); # Peter Luschny, Nov 27 2020

Table[-BitXor[-i,i], {i, 200}] (* Peter Luschny, Jun 01 2011 *)
a[n_] := 2^(IntegerExponent[n, 2] + 1); Array[a, 100] (* Jean-François Alcover, May 09 2017 *)

A171977(n) = 2^(1+valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

def A171977(n): return (n&-n)<<1 # Chai Wah Wu, Jul 13 2022

a(n) = (A000265(2*n+1)-1)/A000265(2*n).
a(n) = -(-n XOR n).  XOR the bitwise operation on the two's complement representation for negative integers. - Peter Luschny, Jun 01 2011
a(n) = A038712(n)+1. - Franz Vrabec, Mar 03 2012
a(n) = 2^A001511(n). - Franz Vrabec, Oct 22 2012
a(n) = A046161(n)/A046161(n-1). - Johannes W. Meijer, Nov 04 2012
a(n) = 2^(1 + (A183063(n)/A001227(n))). - Omar E. Pol, Nov 06 2018
a(n) = 2*A006519(n). - Antti Karttunen, Nov 06 2018

I edited this sequence, based on an email message from the author. - N. J. A. Sloane, Nov 20 2010

Definition simplified by N. J. A. Sloane, Mar 18 2012

A273506 T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.

Original entry on oeis.org

1, -1, 7, 1, -1, 11, -1, 319, -143, 715, 1, -26, 559, -221, 4199, -2, 139, -323, 6137, -2261, 52003, 1, -10897, 135983, -4199, 527459, -52003, 37145, -1, 15409, -317281, 21586489, -52877, 7429, -88711, 1964315, 1, -76, 269123, -100901, 274873, -8671, 227447, -227447, 39803225, -2, 466003, -213739, 522629, -59074189, 226061641, -10690009, 25701511, -42077695, 547010035

Offset: 1

Bradley Klee, May 23 2016

Triangle read by rows ( see examples ). The phase space trajectory of a simple pendulum can be written as (q,p) = (R(Q)cos(Q),R(Q)sin(Q)), with scaled, canonical coordinates q and p. The present triangle and A273507 determine a power / Fourier series of R(Q): R(Q) = sqrt(4 *k) * (1 + sum k^n * (A273506(n,m)/A273507(n,m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. The period of an oscillator can be computed by T(k) = dA/dE, where A is the phase area enclosed by the phase space trajectory of conserved, total energy E. As we choose expansion parameter "k" proportional to E, the series expansion of the complete elliptic integral of the first kind follows from T(k) with very little technical difficulty ( see examples and Mathematica function R2ToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

For some remarks on this pendulum problem and an alternative way to compute a(n,m) / A273507(n,m) using Lagrange inversion see the two W. Lang links. - Wolfdieter Lang, Jun 11 2016

			n/m  1    2     3     4
------------------------------
1  |  1
2  | -1,  7
3  |  1, -1,    11
4  | -1,  319, -143, 715
------------------------------
R2(Q) = sqrt(4 k) (1 + (1/6) cos(Q)^4 k +  (-(1/45) cos(Q)^6 + (7/72) cos(Q)^8) k^2)
R2(Q)^2 = 4 k + (4/3) cos(Q)^4 k^2 + ( -(8/45) cos(Q)^6 + (8/9) cos(Q)^8)k^3 + ...
I2 = (1/(2 Pi)) Int dQ (1/2)R2(Q)^2 = 2 k + (1/4) k^2 + (3/32) k^3 + ...
(2/Pi) K(k) ~ (1/2)d/dk(I2) = 1 + (1/4) k + (9/64) k^2 + ...
From _Wolfdieter Lang_, Jun 11 2016 (Start):
The rational triangle r(n,m) = a(n, m) / A273507(n,m) begins:
n\m   1          2          3         4   ...
1:   1/6
2: -1/45        7/72
3:  1/630      -1/30      11/144
4: -1/14175   319/56700 -143/3240  715/10368
... ,
row n = 5: 1/467775 -26/42525 559/45360 -221/3888 4199/62208,
row 6: -2/42567525 139/2910600 -323/145800 6137/272160 -2261/31104 52003/746496,
row 7: 1/1277025750 -10897/3831077250 135983/471517200 -4199/729000 527459/13996800 -52003/559872 37145/497664,
row 8:
-1/97692469875 15409/114932317500 -317281/10945935000 21586489/20207880000 -52877/4199040 7429/124416 -88711/746496 1964315/23887872.
... (End)

Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.
Wolfdieter Lang, Remarks on this entry and A273507
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Denominators: A273507. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
RCoefficients[n_] :=  With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
    Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
       Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
R2ToEllK[NMax_] := D[Expand[(2)^(-2) ReplaceAll[R[NMax], RRules[NMax]]^2] /. {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0},k]
Flatten[Numerator@RCoefficients[10]]
R2ToEllK[10]

A098597 Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number.

Original entry on oeis.org

1, 1, 1, 5, 7, 21, 33, 429, 715, 2431, 4199, 29393, 52003, 185725, 334305, 9694845, 17678835, 64822395, 119409675, 883631595, 1641030105, 6116566755, 11435320455, 171529806825, 322476036831, 1215486600363, 2295919134019, 17383387729001, 32968493968795, 125280277081421

Offset: 0

Michael Somos, Sep 15 2004

Also numerators of g.f. c(x/2) = (1-sqrt(1-2x))/x where c(x) = g.f. of A000108. - Paul Barry, Sep 04 2007
Also numerator of x(n)=Sum(x(k)*x(n-k-1):0<=kA086117(n). - Reinhard Zumkeller, Feb 06 2008
Also numerator of (1/Pi)*int(x^n*sqrt((1-x)/x), x=0..1). - Groux Roland, Mar 17 2011
The negative of this sequence appears in the A-sequence of the Riordan triangle A084930 as numerators  4, -2, -seq(a(n-1), n >= 2). The denominators look like 1, seq(A120777(n-1), n >= 1). - Wolfdieter Lang, Aug 04 2014
The series of a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 09 2017

			1/(1 + sqrt(1-x)) = 1/2 + 1/8*x + 1/16*x^2 + 5/128*x^3 + 7/256*x^4 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..500
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
T. Copeland, Addendum to Elliptic Lie Triad

Cf. Equals A000265(A000108(n)).

Essentially the absolute values of A002596. Cf. A000108, A001795.

[Numerator(Catalan(n)/2^(2*n+1)):n in [0..30]]; // Vincenzo Librandi, Jan 14 2016

a:= n-> abs(numer(binomial(1/2, n+1))): seq(a(n), n=0..50); # Alois P. Heinz, Apr 10 2009

Table[Numerator[CatalanNumber[n]/2^(2n+1)],{n,0,30}] (* Harvey P. Dale, Jul 27 2011 *)
A098597[n_] := With[{c = CatalanNumber[n]}, c / 2^IntegerExponent[c, 2]];
Table[A098597[n], {n, 0, 29}]  (* Peter Luschny, Apr 16 2024 *)

{a(n) = if( n < 0, 0, numerator(polcoeff(1 / (1 + sqrt(1 - x + x * O(x^n))), n)))};

Numerators of g.f.: 1/(1 + sqrt(1-x)).

a(n) = A000108(n) / 2^A048881(n).

Edited by Ralf Stephan, Dec 28 2004

A273496 Triangle read by rows: coefficients in the expansion cos(x)^n = (1/2)^n * Sum_{k=0..n} T(n,k) * cos(k*x).

Original entry on oeis.org

1, 0, 2, 2, 0, 2, 0, 6, 0, 2, 6, 0, 8, 0, 2, 0, 20, 0, 10, 0, 2, 20, 0, 30, 0, 12, 0, 2, 0, 70, 0, 42, 0, 14, 0, 2, 70, 0, 112, 0, 56, 0, 16, 0, 2, 0, 252, 0, 168, 0, 72, 0, 18, 0, 2, 252, 0, 420, 0, 240, 0, 90, 0, 20, 0, 2

Offset: 0

Bradley Klee, May 23 2016

These coefficients are especially useful when integrating powers of cosine x (see examples).
Nonzero, even elements of the first column are given by A000984; T(2n,0) = binomial(2n,n).
For the rational triangles for even and odd powers of cos(x) see A273167/A273168 and A244420/A244421, respectively. - Wolfdieter Lang, Jun 13 2016
Mathematica needs no TrigReduce to integrate Cos[x]^k. See link. - Zak Seidov, Jun 13 2016

			n/k|  0   1   2   3   4   5   6
-------------------------------
0  |  1
1  |  0   2
2  |  2   0   2
3  |  0   6   0   2
4  |  6   0   8   0   2
5  |  0   20  0   10  0   2
6  |  20  0   30  0   12  0   2
-------------------------------
cos(x)^4 = (1/2)^4 (6 + 8 cos(2x) + 2 cos(4x)).
I4 = Int dx cos(x)^4 = (1/2)^4 Int dx ( 6 + 8 cos(2x) + 2 cos(4x) ) = C + 3/8 x + 1/4 sin(2x) + 1/32 sin(4x).
Over range [0,2Pi], I4 = (3/4) Pi.

Zak Seidov, No Need For TrigReduce

Cf. A000984, A001790, A046161, A038533, A038534, A273506, A273507, A273167, A273168, A244420, A244421.

T[MaxN_] := Function[{n}, With[
       {exp = Expand[Times[ 2^n, TrigReduce[Cos[x]^n]]]},
       Prepend[Coefficient[exp, Cos[# x]] & /@ Range[1, n],
        exp /. {Cos[_] -> 0}]]][#] & /@ Range[0, MaxN];Flatten@T[10]
(* alternate program *)
T2[MaxN_] := Function[{n}, With[{exp = Expand[(Exp[I x] + Exp[-I x])^n]}, Prepend[2 Coefficient[exp, Exp[I # x]] & /@ Range[1, n], exp /. {Exp[] -> 0}]]][#] & /@ Range[0, MaxN]; T2[10] // ColumnForm (* _Bradley Klee, Jun 13 2016 *)

From Robert Israel, May 24 2016: (Start)
T(n,k) = 0 if n-k is odd.
T(n,0) = binomial(n,n/2) if n is even.
T(n,k) = 2*binomial(n,(n-k)/2) otherwise. (End)

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

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A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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A056982 a(n) = 4^A005187(n). The denominators of the Landau constants.

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A060818 a(n) = 2^(n - HammingWeight(n)) = 2^(n - BitCount(n)) = 2^(n - A000120(n)).

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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

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A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).

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A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.