A334907 -id:A334907 - cp's OEIS frontend

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

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

A223549 Triangle T(n,k), read by rows, giving the numerator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <=n.

Original entry on oeis.org

1, 3, 1, 21, 15, 3, 77, 43, 35, 5, 1155, 885, 1095, 315, 35, 4389, 8589, 7161, 777, 693, 63, 33649, 80353, 42245, 12285, 16485, 3003, 231, 129789, 91635, 233001, 170145, 152625, 20889, 6435, 429, 4023459, 3283533, 9804465, 8625375, 9695565, 1772199, 819819, 109395, 6435, 15646785, 58019335, 49782755, 25638305, 69324255, 31726695, 9794785, 245245, 230945, 12155

Offset: 0

Jean-François Alcover, Mar 22 2013

From Petros Hadjicostas, May 21 2020: (Start)
Let P_n(x) = Sum_{k=0..n} (T(n,k)/A223550(n,k))*x^k be the Boros-Moll polynomial. It follows from the theory in Comtet (1967, pp. 81-83 and 85-86) that the polynomial Q_n(x) = 2^n*n!*P_n(x) has integer coefficients and satisfies the recurrence (x-1)*Q_n(x) = 2*(2*n - 1)*(x^2 - 2)*Q_{n-1}(x) + (16*(n-1)^2 - 1)*(x + 1)*Q_{n-2}(x).
We have integral_{y = 0..infinity} dy/(y^4 + 2*x*y + 1)^(n + 1) = Pi * P_n(x)/(2^(n + (3/2)) * (x + 1)^(n + (1/2))) = Pi * Q_n(x)/(2^(2*n + (3/2)) * n! * (x + 1)^(n + (1/2))) for x > -1 and n integer >= 0.
It also follows from the theory in Comtet (1967, pp. 81-83) that g(t) = (sqrt(x + sqrt(x^2 - 1 + t)) - sqrt(x - sqrt(x^2 - 1 + t))) / sqrt((1 - t) * (x^2 - 1 + t)) = Sum_{n >= 0} t^n * P_n(x)/(2^(n - (1/2)) * (x + 1)^(n + (1/2))) for x >= 1 and 0 <= t < 1.
From Comtet's result, we get g(t)^2 = 2*(x - sqrt(1-t))/((1-t) * (x^2 - 1 + t)) = 2/((1-t) * (x + sqrt(1-t))) = Sum_{n >= 0} (Sum_{k=0..n} P_k(x) * P_{n-k}(x)) / (2^(n-1) * (x+1)^(n+1)) * t^n for 0 <= t < 1 and x > 1. (End)

			P_3(x) = 77/16 + 43*x/4 + 35*x^2/4 + 5*x^3/2.
As a result, integral_{y = 0..infinity} dy/(y^4 + 2*x*y + 1)^4 = Pi * P_3(x)/(2^(3 + (3/2)) * (x + 1)^(3 + (1/2))) = Pi * (40*x^3 + 140*x^2 + 172*x + 77)/(32 * sqrt(2*(x + 1)^7)) for x > -1. - _Petros Hadjicostas_, May 22 2020
From _Bruno Berselli_, Mar 22 2013: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:
       1;
       3,     1;
      21,    15,      3;
      77,    43,     35,      5;
    1155,   885,   1095,    315,     35;
    4389,  8589,   7161,    777,    693,    63;
   33649, 80353,  42245,  12285,  16485,  3003,  231;
  129789, 91635, 233001, 170145, 152625, 20889, 6435, 429;
  ... (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.

Cf. A067001, A223550 (denominators), A334907.

/* As triangle: */ [[Numerator(2^(-2*n)*&+[2^j*Binomial(2*n-2*j, n-j)*Binomial(n+j, j)*Binomial(j, k): j in [k..n]]): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 22 2013

t[n_, k_] := 2^(-2*n)*Sum[ 2^j*Binomial[2*n - 2*j, n-j]*Binomial[n+j, j]*Binomial[j, k], {j, k, n}]; Table[t[n, k] // Numerator, {n, 0, 9}, {k, 0, n}] // Flatten

T(n,k)/A223550(n,k) =  2^(-2*n)*Sum_{j=k..n} 2^j*binomial(2*n - 2*j, n - j)*binomial(n + j, j)*binomial(j, k) = 2^(-2*n)*A067001(n,n-k) for n >= 0 and k = 0..n.
P_n(x) = Sum_{k=0..n} (T(n, k)/A223550(n,k))*x^k = ((2*n)!/4^n/(n!)^2)*2F1([-n, n + 1], [1/2 - n], (x + 1)/2).
From Petros Hadjicostas, May 22 2020: (Start)
Recurrence for the polynomial: 4*n*(n - 1)*(x - 1)*P_n(x) = 4*(2*n - 1)*(n - 1)*(x^2 - 2)*P_{n-1}(x) + (16*(n - 1)^2 - 1)*(x + 1)*P_{n-2}(x).
O.g.f. for P_n(x): sqrt((x + 1)/(1 - 2*(x + 1)*w)/(x + sqrt(1 - 2*(x + 1)*w))). [It follows from Comtet's theory and my comments.]
P_n(1) = Sum_{k=0..n} T(n,k)/A223550(n,k) = A334907(n)/(2^n*n!). (End)

Various sections and name edited by Petros Hadjicostas, May 22 2020

A223550 Triangle T(n,k), read by rows, giving the denominator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 22 2013

As Chen and Xia (2009) state, the Boros-Moll polynomial P_n(x) can be viewed as a Jacobi polynomial P_n^{a,b}(x) with a = n + (1/2) and b = -(n + (1/2)). For more information about the relation of this polynomial P_n(x) to the theory in Comtet (1967, pp. 81-83 and 85-86), see my comments for A223549. - Petros Hadjicostas, May 22 2020

			P_3(x) = 77/16 + 43*x/4 + 35*x^2/4 + 5*x^3/2.
From _Bruno Berselli_, Mar 22 2013: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins as follows:
      1;
      2,     1;
      8,     4,    2;
     16,     4,    4,    2;
    128,    32,   32,   16,    8;
    256,   128,   64,    8,   16,    8;
   1024,   512,  128,   32,   64,   32,   16;
   2048,   256,  256,  128,  128,   32,   32,  16;
  32768,  4096, 4096, 2048, 2048,  512,  512, 256, 128;
  65536, 32768, 8192, 2048, 4096, 2048, 1024,  64, 256, 128;
  ... (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.

Cf. A067001, A223549 (numerators), A334907.

/* As triangle: */ [[Denominator(2^(-2*n)*&+[2^j*Binomial(2*n-2*j, n-j)*Binomial(n+j, j)*Binomial(j, k): j in [k..n]]): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 22 2013

t[n_, k_] := 2^(-2*n)*Sum[ 2^j*Binomial[2*n - 2*j, n-j]*Binomial[n+j, j]*Binomial[j, k], {j, k, n}]; Table[t[n, k] // Denominator, {n, 0, 9}, {k, 0, n}] // Flatten

A223549(n,k)/T(n,k) =  2^(-2*n)*Sum_{j=k..n} 2^j*binomial(2*n - 2*j, n - j)*binomial(n + j, j)*binomial(j, k) = 2^(-2*n)*A067001(n,n-k) for n >= 0 and k = 0..n.
P_n(x) = Sum_{k=0..n} (A223549(n,k)/T(n,k))*x^k = ((2*n)!/4^n/(n!)^2)*2F1([-n, n + 1], [1/2 - n], (x + 1)/2).
From Petros Hadjicostas, May 22 2020: (Start)
Recurrence for the polynomial: 4*n*(n - 1)*(x - 1)*P_n(x) = 4*(2*n - 1)*(n - 1)*(x^2 - 2)*P_{n-1}(x) + (16*(n - 1)^2 - 1)*(x + 1)*P_{n-2}(x).
P_n(1) = Sum_{k=0..n} A223549(n,k)/T(n,k) = A334907(n)/(2^n*n!). (End)

Name edited by Petros Hadjicostas, May 22 2020

A063079 Bisection of A001790.

Original entry on oeis.org

1, 5, 63, 429, 12155, 88179, 1300075, 9694845, 583401555, 4418157975, 67282234305, 514589420475, 15801325804719, 121683714103007, 1879204156221315, 14544636039226909, 1804857108504066435

Offset: 1

N. J. A. Sloane, Aug 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Petros Hadjicostas, Proof of the claim A334907(n)/n! = a(n+1)/A060818(n), 2020.

Cf. A001790, A060818, A334907. Other bisection gives A061548.

seq(numer(binomial(2*n-3/2,-1/2)), n=1..20);

Numerator[Binomial[2Range[20]-3/2,-(1/2)]] (* Harvey P. Dale, Feb 27 2012 *)

Numerators of binomial(2*n-3/2, -1/2).

Because A334907(n)/n! = a(n+1)/A060818(n) for n >= 0, the o.g.f. of a(n+1)/A060818(n), for n >= 0, is (sqrt(1 + sqrt(8*s)) - sqrt(1 - sqrt(8*s)))/sqrt(8*s * (1 - 8*s)), which is the e.g.f. of A334907 (see the link above for a proof). - Petros Hadjicostas, May 16 2020

More terms from Vladeta Jovovic, Aug 07 2001

A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.

Original entry on oeis.org

1, 6, 4, 42, 60, 24, 308, 688, 560, 160, 2310, 7080, 8760, 5040, 1120, 17556, 68712, 114576, 99456, 44352, 8064, 134596, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1038312, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720

Offset: 0

R. J. Mathar, Mar 17 2007

The integral N(a;n) = Integral_{x=0..infinity} 1/(x^4 + 2*a*x^2 + 1)^(n+1) has a polynomial representation P_n(a) = 2^(n + 3/2) * (a+1)^(n + 1/2) * N(a;n) / Pi (known as the Boros-Moll polynomial). The table contains the coefficients T(n,l) of P_n(a) = 2^(-2*n)*Sum_{l=0..n} T(n,l)*a^l in row n and column l (with n >= 0 and 0 <= l <= n).

			The table T(n,l) (with rows n >= 0 and columns l = 0..n) starts:
      1;
      6,     4;
     42,    60,     24;
    308,   688,    560,   160;
   2310,  7080,   8760,  5040,  1120;
  17556, 68712, 114576, 99456, 44352, 8064;
  ...
For n = 2, N(a;2) = Integral_{x=0..oo} dx/(x^4 + 2*a*x + 1)^3 = 2^(-2*2)*(Sum_{l=0..2} T(2,l)*a^l) * Pi/(2^(2 + 3/2) * (a + 1)^(2 + 1/2) = (42 + 60*a + 24*a^2) * Pi/(32 * (2*(a+1))^(5/2)) for a > -1. - _Petros Hadjicostas_, May 25 2020

Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Victor H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Victor H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin., no 2 (2007), #4.

Cf. A002458 (row sums), A004982 (column l=0), A059304 (main diagonal), A067001 (rows reversed), A223549, A223550, A334907.

A126936 := proc(m, l)
    add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m):
end:
seq(seq(A126936(m,l), l=0..m), m=0..12); # R. J. Mathar, May 25 2020

t[m_, l_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, l, m}]; Table[t[m, l], {m, 0, 11}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Maple, adapted May 2020 *)

From Petros Hadjicostas, May 25 2020: (Start)
T(n,l) = A067001(n, n-l) = 2^(2*n) * A223549(n,l)/A223550(n,l).
Sum_{l=0..n} T(n,l) = A002458(n) = A334907(n)*2^n/n!.
Bivariate o.g.f.: Sum_{n,l >= 0} T(n,l)*x^n*y^l = sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))). (End)

Corrected by Petros Hadjicostas, May 23 2020

cp's OEIS Frontend

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

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A223549 Triangle T(n,k), read by rows, giving the numerator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <=n.

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A223550 Triangle T(n,k), read by rows, giving the denominator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <= n.

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A063079 Bisection of A001790.

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A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2*a*x^2 + 1)^(n+1) from x = 0 to infinity.

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A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.