A001790 -id:A001790 - cp's OEIS frontend

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

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

A008316 Triangle of coefficients of Legendre polynomials P_n (x).

Original entry on oeis.org

1, 1, -1, 3, -3, 5, 3, -30, 35, 15, -70, 63, -5, 105, -315, 231, -35, 315, -693, 429, 35, -1260, 6930, -12012, 6435, 315, -4620, 18018, -25740, 12155, -63, 3465, -30030, 90090, -109395, 46189, -693, 15015, -90090, 218790, -230945, 88179, 231, -18018, 225225, -1021020, 2078505, -1939938, 676039

Offset: 0

N. J. A. Sloane

			Triangle starts:
   1;
   1;
  -1,   3;
  -3,   5;
   3, -30, 35;
  15, -70, 63;
  ...
P_5(x) = (15*x - 70*x^3 + 63*x^5)/8 so T(5, ) = (15, -70, 63). P_6(x) = (-5 + 105*x^2 - 315*x^4 + 231*x^6)/16 so T(6, ) = (-5, 105, -315, 231). - _Michael Somos_, Oct 24 2002

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.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
T. Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera, see the Additional Notes section, 2015.
H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Eric Weisstein's World of Mathematics, Legendre Polynomial

Cf. A001790, A001800, A001801.
With zeros: A100258.
Cf. A121448.

Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]

{T(n, k) = if( n<0, 0, polcoeff( pollegendre(n) * 2^valuation( (n\2*2)!, 2), n%2 + 2*k))}; /* Michael Somos, Oct 24 2002 */

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

A038534 Numerators of coefficients of EllipticK/Pi.

Original entry on oeis.org

1, 1, 9, 25, 1225, 3969, 53361, 184041, 41409225, 147744025, 2133423721, 7775536041, 457028729521, 1690195005625, 25145962430625, 93990019574025, 90324408810638025, 340357374376418025, 5147380044581630625, 19520119892056100625, 1187604094232693162025

Offset: 0

Wouter Meeussen, revised Jan 03 2001

The denominators are given in A038533.
Also numerators in expansion of the hypergeometric series 2F1(1/2,1/2; 1; x).
This means numerators of the expansion coefficients of 2*K(k)/Pi = 2F1(1/2,1/2; 1; k^2) in powers of k^2, with K(k) the complete elliptic integral of the first kind. The denominators are given in A056982. The period T of the plane pendulum (mass m, length L, Earth's gravity g, energy E) is 4*sqrt(L/g)*K(sin(phi_0/2)) with cos(phi_0) = -E/(m*g*L) (maximal phi value). See the Landau - Lifschitz reference, p. 30. - Wolfdieter Lang, May 29 2016
It is easy and inexpensive to make a satisfactory precision measurement of a(1)/4, a(2)/64, and a(3)/256 using a pendulum rigged from a computer mouse. In "Digital Pendulum Data Analysis" (see links) amplitude vs. time data is transformed to period vs. sin(phi_0/2)^2 data, thus allowing extraction of expansion coefficients as fit parameters. - Bradley Klee, Dec 25 2016

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 91, Eq. 2.1.
L. D. Landau und E. M. Lifschitz, Mechanik, Akademie Verlag, Berlin, 1967, p. 30 (Exercise 1 in chapter III, paragraph 11.)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Klee, Digital Pendulum Data Analysis: Output, Github, 2016.
David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, arXiv:2109.00027 [math.AG], 2021. See (1.2) p. 1.
G. N. Watson, A Note on Gamma Functions.Edinburgh Mathematical Notes, 42, 1959, pp 7-9.

Cf. A001790, A038533, A038535, A056040, A056982, A060632.

swing := proc(n) option remember; if n = 0 then 1 elif n mod 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))^2; seq(a(n),n=0..20); # Peter Luschny, Aug 06 2014

Numerator@ CoefficientList[ Series[ EllipticK@x, {x, 0, 19}]/Pi, x] (* Robert G. Wilson v, Jul 19 2007 *)

a(n) = 2^(-2*w(n))*binomial(2*n,n)^2 with w(n) = A000120(n), the number of 1's in binary expansion of n.
a(n) = A001790(n)^2.
a(n) = (A056040(2*n)/A060632(2*n))^2. - Peter Luschny, Aug 06 2014
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n-1/2,n). - Peter Luschny, Apr 08 2016
a(n) = numerator(((2*n)!/(2^(2*n)*(n!)^2))^2). - Stefano Spezia, May 01 2025

A046643 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 35, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 63, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 35, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 231, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 35, 35, 1, 1, 3, 1, 1, 1, 5, 1, 3

Offset: 1

N. J. A. Sloane, Dec 11 1999

b(n) = A046643(n)/A046644(n) is multiplicative with b(p^n) = (2n-1)!!/2^n/n!. Dirichlet g.f. of A046643(n)/A046644(n) is sqrt(zeta(x)). - Christian G. Bower, May 16 2005

That is, b(p^n) = A001147(n) / (A000079(n)*A000142(n)) = A010050(n)/A000290(A000165(n)) = (2n)!/((2^n*n!)^2). - Antti Karttunen, Jul 08 2017

			b_1, b_2, ... = 1, 1/2, 1/2, 3/8, 1/2, 1/4, 1/2, 5/16, 3/8, 1/4, 1/2, 3/16, ...

Cf. A000079, A000165, A001147, A001790, A005187, A010050, A028234, A067029.

Cf. A046644, A046645.

b := proc(n) option remember; local c,i,t1; if n = 1 then 1 else c := 1; t1 := divisors(n);
for i from 2 to nops(t1)-1 do c := c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;

b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
Do[c = c - b[dn[[i]]]*b[n/dn[[i]]], {i, 2, Length[dn] - 1}]; c/2); a[n_] := Numerator[b[n]]; a /@ Range[90] (* Jean-François Alcover, Apr 04 2011, after Maple version *)

A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
A046643(n) = numerator(A046643perA046644(n)); \\ After Maple-program, Antti Karttunen, Jul 08 2017

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

(define (A046643 n) (if (= 1 n) n (* (A001790 (A067029 n)) (A046643 (A028234 n)))))
;; Or, after Christian G. Bower's May 16 2005 comment:
(definec (A046643perA046644 n) (if (= 1 n) n (* (/ (A010050 (A067029 n)) (A000290 (A000165 (A067029 n)))) (A046643perA046644 (A028234 n)))))
(define (A046643 n) (numerator (A046643perA046644 n)))
(define (A046644 n) (denominator (A046643perA046644 n)))
;; Antti Karttunen, Jul 08 2017

Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].
From Antti Karttunen, Jul 08 2017: (Start)
Multiplicative with a(p^n) = A001790(n).
a(1) = 1; for n > 1, a(n) = A001790(A067029(n)) * a(A028234(n)).
(End)
Sum_{j=1..n} A046643(j)/A046644(j) ~ n / sqrt(Pi*log(n)) * (1 + (1 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 04 2025

A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.

Original entry on oeis.org

1, 0, 1, -1, 0, 3, 0, -3, 0, 5, 3, 0, -30, 0, 35, 0, 15, 0, -70, 0, 63, -5, 0, 105, 0, -315, 0, 231, 0, -35, 0, 315, 0, -693, 0, 429, 35, 0, -1260, 0, 6930, 0, -12012, 0, 6435, 0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155, -63, 0, 3465, 0, -30030, 0, 90090, 0, -109395, 0, 46189

Offset: 0

Ralf Stephan, Nov 13 2004

For a relation to Jacobi quartic elliptic curves, see the MathOverflow link. For a self-convolution of the polynomials relating them to the Chebyshev and Fibonacci polynomials, see A049310 and A053117. For congruences and connections to other polynomials (Jacobi, Gegenbauer, and Chebyshev) see the Allouche et al. link. For relations to elliptic cohomology and modular forms, see references in Copeland link.- Tom Copeland, Feb 04 2016

			Triangle begins:
   1;
   0,   1;
  -1,   0,     3;
   0,  -3,     0,   5;
   3,   0,   -30,   0,   35;
   0,  15,     0, -70,    0,   63;
  -5,   0,   105,   0, -315,    0,    231;
   0, -35,     0, 315,    0, -693,      0, 429;
  35,   0, -1260,   0, 6930,    0, -12012,   0, 6435;
  ...

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.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, p. 21-49.
Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera
H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.

Without zeros: A008316. Row sums are A060818.
Columns (with interleaved zeros and signs) include A001790, A001803, A100259. Diagonals include A001790, A001800, A001801, A001802.
Cf. A049310, A005187, A049600, A053117.

row[n_] := CoefficientList[ LegendreP[n, x], x]*2^IntegerExponent[n!, 2]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)

a(k,n)=polcoeff(pollegendre(k,x),n)*2^valuation(k!,2)

from mpmath import *
mp.dps=20
def a007814(n):
    return 1 + bin(n - 1)[2:].count('1') - bin(n)[2:].count('1')
for n in range(11):
    y=2**sum(a007814(i) for i in range(2, n+1))
    l=chop(taylor(lambda x: legendre(n, x), 0, n))
    print([int(i*y) for i in l]) # Indranil Ghosh, Jul 02 2017

The n-th normalized Legendre polynomial is generated by 2^(-n-a(n)) (d/dx)^n (x^2-1)^n / n! with a(n) = A005187(n/2) for n even and a(n) = A005187((n-1)/2) for n odd. The non-normalized polynomials have the o.g.f. 1 / sqrt(1 - 2xz + z^2). - Tom Copeland, Feb 07 2016

The consecutive nonzero entries in the m-th row are, in order, (c+b)!/(c!(m-b)!(2b-m)!*A048896(m-1)) with sign (-1)^b where c = m/2-1, m/2, m/2+1, ..., (m-1) and b = c+1 if m is even and sign (-1)^c with c = (m-1)/2, (m-1)/2+1, (m-1)/2+2, ..., (m-1) with b = c+1 if m is odd. For the 9th row the 5 consecutive nonzero entries are 315, -4620, 18018, -25740, 12155 given by c = 4,5,6,7,8 and b = 5,6,7,8,9. - Richard Turk, Aug 22 2017

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

A038003 Odd Catalan numbers: a(n) = A000108(2^n-1).

Original entry on oeis.org

1, 1, 5, 429, 9694845, 14544636039226909, 94295850558771979787935384946380125, 11311095732253345760960290897769189975961199415637572612957718759342193629

Offset: 0

Christian G. Bower

nonn

The next term has 150 digits. - Harvey P. Dale, Feb 22 2016

David Wasserman, May 07 2007, Table of n, a(n) for n = 0..9
H-Y. Lin, Odd Catalan Numbers modulo 2^k, Integers 11 (2011) #A55
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A094389, A119861, A119908, A120274, A120275.

Intersection of A001790 and A098597. - Dimitri Papadopoulos, Oct 28 2016

[Binomial(2^(n+1)-2, 2^n-1)/(2^n): n in [0..10]]; // Vincenzo Librandi, Nov 01 2016

Select[CatalanNumber[Range[0,300]],OddQ] (* Harvey P. Dale, Feb 22 2016 *)

a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n); \\ Joerg Arndt, Nov 05 2015

from _future_ import division
A038003_list, c, s = [1, 1], 1, 3
for n in range(2,10**5+1):
    c = (c*(4*n-2))//(n+1)
    if n == s:
        A038003_list.append(c)
        s = 2*s+1 # Chai Wah Wu, Feb 12 2015

a(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n).
a(n-1) = C(2^n,2^(n-1))/(2^n - 1)/2. - Benoit Cloitre, Aug 17 2002
a(n) = A000108(2^n-1). - David Wasserman, May 07 2007

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]

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)

cp's OEIS Frontend

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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A008316 Triangle of coefficients of Legendre polynomials P_n (x).

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A038534 Numerators of coefficients of EllipticK/Pi.

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A046643 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.

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A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.

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