A049606 -id:A049606 - cp's OEIS frontend

A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 0, 2, 2, 1, 1, 2, 2, 0, 0, 0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0

Offset: 0

Claude Lenormand (hlne.lenormand(AT)voono.net), Nov 05 2001

T(m,k) is the number of 'carries' that occur when adding k and m-k in base 2 using the traditional addition algorithm. - Tom Edgar, Jun 10 2014

			Triangle begins:
[0]
[0, 0]
[0, 1, 0]
[0, 0, 0, 0]
[0, 2, 1, 2, 0]
[0, 0, 1, 1, 0, 0]
[0, 1, 0, 2, 0, 1, 0]
[0, 0, 0, 0, 0, 0, 0, 0]
[0, 3, 2, 3, 1, 3, 2, 3, 0]
[0, 0, 2, 2, 1, 1, 2, 2, 0, 0]
[0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0]
... - _N. J. A. Sloane_, Aug 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..10010; the first 141 antidiagonals, flattened
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Row sums: A187059.

Cf. A007318, A007814, A001511, A000120, A047999, A049606, A000680, A048881, A011371, A005187, A000265, A001316, A001317, A243759.

A065040 := (n, k) -> padic[ordp](binomial(n, k), 2):
seq(seq(A065040(n,k), k=0..n), n=0..13); # Peter Luschny, Aug 15 2017

T[m_, k_] := IntegerExponent[Binomial[m, k], 2]; Table[T[m, k], {m, 0, 13}, {k, 0, m}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

T(m,k)=hammingweight(k)+hammingweight(m-k)-hammingweight(m)
for(m=0,9,for(k=0,m,print1(T(m,k)", "))) \\ Charles R Greathouse IV, Mar 26 2013

As an array f(i,j) = f(j,i) = T(i+j,j) read by antidiagonals: f(0,j) = 0, f(1,j) = A007814(j+1), f(i,j) = Sum_{k=0..i-1} (f(1,j+k) - f(1,k)). [corrected by Kevin Ryde, Oct 07 2021]
The n-th term a(n) is equal to the binomial coefficient binomial(m,k), where m = floor((1+sqrt(8*n+1))/2) - 1 and k = n - m(m+1)/2. Also a(n) = g(m) - g(k) - g(m-k), where g(x) = Sum_{i=1..floor(log_2(x))} floor(x/2^i), m = floor((1+sqrt(8*n+1))/2) - 1, k = n - m(m+1)/2. - Hieronymus Fischer, May 05 2007
T(m,k) <= log_2 m, for m > 0. - Charles R Greathouse IV, Mar 26 2013
T(m,k) = log_2(A082907(m,k)). - Tom Edgar, Jun 10 2014
From Antti Karttunen, Oct 28 2014: (Start)
a(n) = A007814(A007318(n)).
a(n) * A047999(n) = 0 and a(n) + A047999(n) > 0 for all n.
(End)

Name clarified by Antti Karttunen, Oct 28 2014

A067667 a(n) = (2^n)!/2^(2^n-1).

Original entry on oeis.org

1, 1, 3, 315, 638512875, 122529844256906551386796875, 13757108753595648665519665029568345104465749222289382342659100341796875

Offset: 0

Benoit Cloitre, Feb 04 2002

nonn

a(n) is also the number of knockout tournament seedings with 2^n teams. - Alexander Karpov, Aug 09 2015
From Zhujun Zhang, Jun 17 2019: (Start)
a(n) is also the number of heap-ordered binomial trees of order n (i.e., binomial heaps with 2^n nodes), see the Mark R. Brown reference.
a(n) is also the largest odd divisor of (2^n)!. (End)

Jianing Song, Table of n, a(n) for n = 0..8
Mark R. Brown, Implementation and analysis of binomial queue algorithms, SIAM Journal on Computing, 1978, 7(3):298-319.
Alexander Karpov, A theory of knockout tournament seedings, Heidelberg University, AWI Discussion Paper Series, No. 600.
Zhujun Zhang, A Note on Counting Binomial Heaps, ResearchGate, June 2019.

Cf. A049606, A069954.

[Factorial(2^n)/2^(2^n-1): n in [1..6]]; // Vincenzo Librandi, Aug 10 2015

Table[(2^n)! / 2^(2^n - 1), {n, 6}] (* Vincenzo Librandi, Aug 10 2015 *)

a(n) = (2^n)!/2^(2^n-1) \\ Jianing Song, Jul 15 2021

From Alexander Karpov, Aug 09 2015: (Start)
a(n) = (2^n)!/2^(2^n-1).
a(n) = (2^n-1)!!*a(n-1).
a(n) = binomial(2^n-1, 2^(n-1)-1)*(a(n-1))^2 = A069954(n-1) * (a(n-1))^2.
(End)
a(n) = A049606(2^n). - Zhujun Zhang, Jun 16 2019
a(n) = Product_{odd k < 2^n} k^(n - floor(log_2(k))). - Harry Richman, May 18 2023

a(0) prepended by Jianing Song, Jul 15 2021

A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 45, 40, 3, 6, 1, 1, 1, 40320, 315, 80, 15, 24, 1, 2, 1, 1, 362880, 315, 560, 45, 24, 1, 6, 1, 1, 1, 3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1

Offset: 0

Paul Curtz, Dec 10 2008

a(n) is the last sequence of a trio with, first, A141412 and, second, A142048 (denominators).

			Contribution from _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
        1,
        1,    1,
        2,    1,    1,
        6,    1,    1,   1,
       24,    3,    2,   1,   1,
      120,    3,    2,   1,   1, 1,
      720,   15,    8,   3,   2, 1,  1,
     5040,   45,   40,   3,   6, 1,  1, 1,
    40320,  315,   80,  15,  24, 1,  2, 1, 1,
   362880,  315,  560,  45,  24, 1,  6, 1, 1, 1,
  3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1,
  ...
First column: A000142; second column: A049606. (End)

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A152650, A142048, A142412,

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Denominator (* Jean-François Alcover, Oct 02 2012 *)

More terms from Jean-François Alcover, Oct 02 2012

A004130 Numerators in expansion of (1-x)^{-1/4}.

Original entry on oeis.org

1, 1, 5, 15, 195, 663, 4641, 16575, 480675, 1762475, 13042315, 48612265, 729183975, 2748462675, 20809788825, 79077197535, 4823709049635, 18443593425075, 141400882925575, 543277076503525, 8366466978154285, 32270658344309385

Offset: 0

N. J. A. Sloane

Numerators in expansion of sqrt(1/sqrt(1-4x)). - Paul Barry, Jul 12 2005

Denominators are in A088802. - Michael Somos, Aug 23 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A004134, A004981, A034255, A034385, A048882, A007696, A000265, A049606.

Table[Numerator[Binomial[-1/4, n] (-1)^n], {n, 0, 20}]

{a(n) = if( n<0, 0, numerator( polcoeff( (1 - x +x*O(x^n))^(-1/4), n ) ) ) } /* Michael Somos, Aug 23 2007 */

a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by Mitch Harris, following a conjecture by Ralf Stephan.
a(n) = 2^(e_2((2n)!)-n)/n! Product[4k+1,{k,0,n-1}], where e_2((2n)!) is the highest power of 2 that divides (2n)! (sequence A005187). - Emanuele Munarini, Jan 25 2011
Numerators in (1-4t)^(-1/4) = 1 + t + (5/2)t^2 + (15/2)t^3 + (195/8)t^4 + (663/8)t^5 + (4641/16)t^6 + (16575/16)t^7 + ... = 1 + t + 5*t^2/2! + 45*t^3/3! + 585*t^4/4! + ... = e.g.f. for the quartic factorials A007696 (cf. A094638). - Tom Copeland, Dec 04 2013

A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.

Original entry on oeis.org

1, 3, 315, 155925, 212837625, 618718975875, 3287253918823875, 28845653137679503125, 388983632561608099640625, 7637693625347175036443671875, 209402646126143497974176151796875, 7752714167528210725497923667975703125, 377130780679409810741846496828678078515625

Offset: 0

Reinhard Zumkeller, Apr 25 2008

nonn

From Karol A. Penson, Oct 05 2009: (Start)
Integral representation as n-th moment of a positive function on a positive semi-axis (solution of the Stieltjes moment problem), in Maple notation:
a(n)=int(x^n*((1/4)*sqrt(2)*(Pi^(3/2)*2^(1/4)*hypergeom([], [1/2, 3/4], -(1/32)*x)*sqrt(x)-2*Pi*hypergeom([], [3/4, 5/4], -(1/32)*x)*GAMMA(3/4)*x^(3/4)+sqrt(Pi)*GAMMA(3/4)^2*2^(1/4)*hypergeom([], [5/4, 3/2],-(1/32)*x)*x)/(Pi^(3/2)*GAMMA(3/4)*x^(5/4))), x=0..infinity), n=0,1... .
This solution may not be unique. (End)

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Appendix: Problem 203.1, p164.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Tournament
Index entries for sequences related to tornaments.

Cf. A008299, A000142, A100733, A001018.

a(n)={(4*n)!/(n!*8^n)} \\ Andrew Howroyd, Jan 07 2020

a(n) = (4*n)!/(n!*8^n).
a(n) = A001147(n)*A001147(2*n).
a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - Reinhard Zumkeller, Apr 28 2008

Terms a(11) and beyond from Andrew Howroyd, Jan 07 2020

A160624 Denominator of Laguerre(n, 2).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 638512875, 58046625, 10854718875, 8881133625, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 3792985290950625, 147926426347074375

Offset: 0

N. J. A. Sloane, Nov 14 2009

			1, -1, -1, -1/3, 1/3, 11/15, 37/45, 209/315, 113/315, 23/2835, -4381/14175, -84389/155925, -310517/467775, ... = A160623/A160624.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

For numerators see A160623. Different from A049606.

Cf. A295382.

[Denominator((&+[Binomial(n,k)*((-2)^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018

seq(denom(orthopoly[L](n,2)), n=0 .. 100); # Robert Israel, Jul 23 2015

Denominator[LaguerreL[Range[0,30],2]] (* Vincenzo Librandi, May 24 2012 *)

for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*((-2)^k/k!))), ", ")) \\ G. C. Greubel, May 06 2018

a(n) = denominator(pollaguerre(n, 0, 2)); \\ Michel Marcus, Feb 05 2021

Denominators of coefficients in expansion of exp(-2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Aug 29 2018

A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 5, 1, 1, 3, 1, 2, 1, 13, 1, 2, 1, 7, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 9, 1, 1, 3, 19, 1, 1, 1, 5, 1, 1, 1, 21, 1

Offset: 1

Amiram Eldar, Sep 03 2024

The product of the prime powers in the prime factorization of n that have an exponent that is smaller than the maximum exponent in this factorization.

			60 = 2^2 * 3 * 5, and the maximum exponent in the prime factorization of 60 is 2, which is the exponent of its prime factor 2. Therefore a(60) = 3 * 5 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A049606, A051903, A072774, A375931, A375933.

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; n / (Times @@ p[[i]])^m]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] < m, p[i]^e[i], 1)));}

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i < max_{j} e_j]), where [] is the Iverson bracket.
a(n) = n / A375931(n).
a(n) = 1 if and only if n is a power of a squarefree number (A072774).
A051903(a(n)) = A375933(n).
a(n!) = A049606(n) for n != 3.

A067655 Denominators of the coefficients in exp(2x/(1-x)) power series.

Original entry on oeis.org

1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 3869775, 638512875, 638512875, 10854718875, 8881133625, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 2595200462229375

Offset: 1

Benoit Cloitre, Feb 03 2002

Differs from A049606 at positions enumerated by A097324.

K. Knopp, Theory and application of infinite series, Dover, p. 547.
O. Perron, Über das infinitäre Verhalten der Koeffizienten einer gewissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 329-340, 1914.

Cf. A067654.

a(n) is the denominator in Sum_{i=1..n} binomial(n-1, i-1)*2^i/i!.

A095987 a(n) = gcd(n!!, (n-1)!!) where n!! = A006882.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 15, 15, 45, 45, 315, 315, 315, 315, 2835, 2835, 14175, 14175, 155925, 155925, 467775, 467775, 6081075, 6081075, 42567525, 42567525, 638512875, 638512875, 638512875, 638512875, 10854718875, 10854718875, 97692469875, 97692469875

Offset: 0

Leroy Quet, Jul 18 2004

nonn

Let f_n(m) be a multifactorial: for m = positive integer, f_n(m) = Product_{k=0..floor((m-1)/n)} (m - k*n). E.g., f_2(m) = m!!. f_n(0) is defined as 1.

a(2n) gives A049606.

a:= n-> (d-> gcd(d(n), d(n-1)))(doublefactorial):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 26 2019

f[n_] := GCD[n!!, (n - 1)!! ]; Table[ f[n], {n, 35}]
GCD@@#&/@Partition[Range[0,40]!!,2,1] (* Harvey P. Dale, May 04 2015 *)

a(2m) = a(2m+1) = A049606(m).

Edited and extended by Robert G. Wilson v, Jul 19 2004

Missing a(0)=1 inserted by Alois P. Heinz, Oct 26 2019

A123746 Numerators of partial sums of a series for 1/sqrt(2).

Original entry on oeis.org

1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 53747636888759

Offset: 0

Wolfdieter Lang, Nov 10 2006

Denominators are given by A046161(n),n>=0.
The alternating sum over central binomial coefficients scaled by powers of 4, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/4^k, has the limit s = lim_{n->infinity} r(n) = 1/sqrt(2). From the expansion of 1/sqrt(1-x) for |x|<1 which extends to x=-1 due to Abel's limit theorem and the convergence of the series s. See the W. Lang link.
(2^n)*n!*r(n) = A003148(n). - Wolfdieter Lang, Oct 06 2008

			a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Rationals and more.
Michael Milgram, An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function, ResearchGate, 2024. See p. 20.

Cf. A120088/(2*A120777) partial sums for a series of sqrt(2).

Equals A003148 divided by A049606. - Johannes W. Meijer, Nov 23 2009

List([0..30], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/(-4)^k )) ); # G. C. Greubel, Aug 10 2019

[Numerator( (&+[Binomial(2*k,k)/(-4)^k: k in [0..n]])): n in [0..30]]; // G. C. Greubel, Aug 10 2019

A123746:=n-> numer(add(binomial(2*k,k)/(-4)^k, k=0..n)); seq(A123746(n), n=0..30); # G. C. Greubel, Aug 10 2019
a := n -> numer(add(binomial(-1/2, j), j=0..n));
seq(a(n), n=0..24); # Peter Luschny, Sep 26 2019

Table[Numerator[Sum[Binomial[2*k, k]/(-4)^k, {k,0,n}]], {n,0,30}] (* G. C. Greubel, Mar 28 2018 *)

{r(n) = sum(k=0,n,(-1/4)^k*binomial(2*k,k))};
vector(30, n, n--; numerator(r(n)) ) \\ G. C. Greubel, Mar 28 2018

[numerator( sum(binomial(2*k,k)/(-4)^k for k in (0..n)) ) for n in (0..30)] # G. C. Greubel, Aug 10 2019

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k* binomial(2*k,k)/4^k, n>=0.
r(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!/(2*k)!!, n>=0, with the double factorials A001147 and A000165.
r(n) = 1/sqrt(2) - binomial(-1/2, 1 + n)*hypergeom([1, 3/2 + n], [2 + n], -1). - Peter Luschny, Sep 26 2019

cp's OEIS Frontend

A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).

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A067667 a(n) = (2^n)!/2^(2^n-1).

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A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

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A004130 Numerators in expansion of (1-x)^{-1/4}.

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A139541 There are 4*n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4*n)!/(n!*8^n) different ways.

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A160624 Denominator of Laguerre(n, 2).

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A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

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A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.