A056040 -id:A056040 - cp's OEIS frontend

A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

1, 17, 69, 1113, 17817, 285297, 1141213, 18260633, 1168681737, 18699007017, 74796032037, 1196736992841, 19147791938817, 306364680039081, 1225458720340365, 19607339566855065, 5019478929156305865, 80311662878468159865, 321246651514020383485, 5139946424277661728785

Offset: 0

Peter Luschny, Sep 27 2019

This sequence is a variant of the Landau constants when the normalized central binomial is replaced by the normalized swinging factorial.
(1) A277233(n)/4^A005187(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2 with the normalized central binomial
    g(n) = (2*n)! / (2^n*n!)^2 = A001790(n)/A046161(n).
(2) A327495(n)/4^A327492(n) are the rationals considered here. These numbers are defined as H(n) = Sum_{j=0..n} h(j)^2 with the normalized swinging factorial
    h(n) = n! / (2^n*floor(n/2)!)^2 = A163590(n)/A327493(n).
(3) In particular, this means that we have the pure integer representations
    A277233(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2;
    A327495(n) = Sum_{k=0..n}(A163590(k)*(2^(A327492(n) - A327492(k))))^2.
(4) A163590 is the odd part of the swinging factorial and A001790 is the odd part of the swinging factorial at even indices (see a comment from Aug 01 2009 in A001790). Similarly, A327493(2n)=A046161(2n) and A327493(2n+1) = 2*A046161(2n+1).
(5) A005187 are the partial sums of A001511, the 2-adic valuation of 2n, and A327492 are the partial sums of A327491.

			r(n) = 1, 17/16, 69/64, 1113/1024, 17817/16384, 285297/262144, 1141213/1048576, 18260633/16777216, ...

Denominators in A327496.
Cf. A327491, A327492, A327493, A327494.
Cf. A277233, A005187, A056040, A000984, A001790/A046161, A163590, A001511.

A327495 := n -> numer(add(j!^2/(2^j*iquo(j,2)!)^4, j=0..n)):
seq(A327495(n), n=0..19);

a(n)={ numerator(sum(j=0, n, (j!/(2^j*(j\2)!)^2)^2 )) } \\ Andrew Howroyd, Sep 28 2019

Denominator(r(n)) = 4^A327492(n) = A327493(n)^2 = A327496(n).

a(n) = Sum_{k=0..n} (A163590(k)*(2^(A327492(n) - A327492(k))))^2.

A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

Original entry on oeis.org

1, 5, 19, 67, 227, 751, 2445, 7869, 25107, 79567, 250793, 786985, 2460397, 7667921, 23832931, 73902627, 228692115, 706407903, 2178511449, 6708684009, 20632428249, 63380014845, 194486530791, 596213956023, 1826103432573, 5588435470401, 17089296473655

Offset: 0

Peter Luschny, Aug 06 2009

nonn

Also a(n) = sum {i=0..n} (-1)^(n-i) binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163772.

a := proc(n) local i; add((-1)^(n-i)*binomial(n,i)/Beta(i+1,i+1),i=0..n) end:
seq(simplify((-1)^n*hypergeom([-n,3/2], [1], 4)),n=0..26); # Peter Luschny, Apr 26 2016

CoefficientList[Series[Sqrt[x+1]/(1-3*x)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[(-1)^(n-i)*Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)

O.g.f.: A(x)=1/(1-x*M(x))^3, M(x) - o.g.f. of A001006. a(n) = sum(k^3/n *sum(C(n,j)*C(j,2*j-n-k), j=0..n), k=1..n). - Vladimir Kruchinin, Sep 06 2010
Recurrence: n*a(n) = (2*n+3)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 21 2012
a(n) ~ 4*3^(n-1/2)*sqrt(n)/sqrt(Pi). - Vaclav Kotesovec, Oct 21 2012
a(n) = (-1)^n*hypergeom([-n,3/2], [1], 4). - Peter Luschny, Apr 26 2016
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} (-3)^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)

A181860 a(n) = lcm(n^2, swinging_factorial(n)).

Original entry on oeis.org

0, 1, 4, 18, 48, 150, 180, 980, 2240, 5670, 6300, 30492, 11088, 156156, 168168, 257400, 1647360, 3719430, 3938220, 17551820, 18475600, 81477396, 85357272, 373173528, 389398464, 1690097500, 1757701400, 7582037400, 3931426800, 33738060600

Offset: 0

Peter Luschny, Nov 21 2010

nonn

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

Cf. A000290, A056040, A170825, A181861.

A181860 := n -> ilcm(n^2,n!/iquo(n,2)!^2);

Join[{0},sf[n_]:=n!/Quotient[n, 2]!^2; a[n_]:=LCM[n^2, sf[n]]; Table[a[n], {n, 30}] ] (* Jean-François Alcover, Jun 28 2013 *)

a(n) = lcm(n^2, n!/(n\2)!^2); \\ Michel Marcus, Mar 06 2018

a(n) = lcm(A000290(n), A056040(n)).

a(26)-a(29) from Vincenzo Librandi, Mar 05 2018

A189911 Row sums of the extended Catalan triangle A189231.

Original entry on oeis.org

1, 2, 4, 9, 18, 40, 80, 175, 350, 756, 1512, 3234, 6468, 13728, 27456, 57915, 115830, 243100, 486200, 1016158, 2032316, 4232592, 8465184, 17577014, 35154028, 72804200, 145608400, 300874500, 601749000, 1240940160, 2481880320, 5109183315, 10218366630

Offset: 0

Peter Luschny, May 01 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.

Cf. A037965, A097070, A056040, A212303.

A189911 := proc(n) local a,b,d; if n = 0 then 1 else
a := GAMMA(n-floor(n/2)); b := GAMMA(floor(n/2+3/2));
d := GAMMA(floor(n/2+1))^2; GAMMA(n+1)*(a*b+d)/(a*b*d) fi end: seq(A189911(n),n=0..32);
A189911 := proc(n) h:=irem(n,2); g:=iquo(n,2); (g+h+1)*binomial(2*g+h,g+h) end; # Peter Luschny, Oct 24 2013

a[n_] := Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; (q+r+1)*Pochhammer[q+1, q+r]/(q+r)!]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jan 09 2014 *)

def A189911():
    r, n = 1, 1
    while True:
        yield r
        h = n//2
        r *= 2 if is_even(n) else (h+2)*(2*h+1)/(h+1)^2
        n += 1
a = A189911(); [next(a) for i in range(16)]  # Peter Luschny, Oct 24 2013

Let a = Gamma(n-floor(n/2)), b = Gamma(floor(n/2+3/2)), d = Gamma( floor(n/2+1))^2, c = Gamma(n+1). Then a(n) = c*(a*b+d)/(a*b*d).
a(n) = A162246(n,n) + A162246(n,n+1) for n > 0.
From Peter Luschny, Oct 24 2013 : (Start)
E.g.f.: (x+1)*(BesselI(0, 2*x)+BesselI(1, 2*x)).
O.g.f.: I*(2*x^2-1)/(2*sqrt(2*x+1)*x*(2*x-1)^(3/2))-1/(2*x).
Recurrence: a(0) = 1; a(n) = a(n-1)*2 if n is even else ([n/2]+2)*(2*[n/2]+1)/([n/2]+1)^2. ([.] the floor brackets.)
a(n) = A056040(n) + A212303(n) = n$*(1+[(n+1)/2]^((-1)^n)), where n$ is the swinging factorial.
a(2*n) = (n+1)*C(2*n, n) (A037965);
a(2*n+1) = (n+2)*C(2*n+1, n+1) (A097070). (End)
Sum_{n>=0} 1/a(n) = 4*Pi/sqrt(3) - Pi^2/3 - 2. - Amiram Eldar, Aug 20 2022
D-finite with recurrence: (n-2)*(n+1)^2*a(n) - (2*(n-2)^2+2*n-12)*a(n-1) - 4*(n+2)*(n-1)^2*a(n-2) = 0. - Georg Fischer, Nov 25 2022

A356637 a(n) = A000265(A263931(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 3, 3, 45, 5, 1, 21, 7, 175, 675, 45, 45, 1485, 5775, 5775, 45045, 2145, 195, 8775, 2925, 5733, 22491, 833, 6545, 373065, 24871, 24871, 1566873, 3086265, 181545, 357903, 39767, 39767, 156975, 309925, 61985, 5020785, 239085, 20322225, 160730325

Offset: 0

Peter Luschny, Sep 07 2022

nonn

Let n >= 5. If a(n) is squarefree, then 2 divides binomial(2*n, n) more than once and is the only prime that does so. There is only a finite number of such cases (see A059097).
An efficient algorithm for the calculation is available, which is based on prime factorization. See the SageMath implementation. The main application is the efficient calculation of the central binomial coefficient, which is the product of this sequence, the Glaisher/Gould sequence, and the upper primorial function (see the formula section).
Since the central binomial coefficient is a bisection of the swinging factorial A056040, and the swinging factorial, in turn, is the building block for an efficient algorithm for the computation of the factorial function, the terms of this sequence occur as factors in all these computations. See the links for details.

			Let n = 22 and consider the prime factorization of m = binomial(2*n, n):
2^3 * [3 * 5 * 13] * 23 * 29 * 31 * 37 * 41 * 43. Then a(22) = 3 * 5 * 13. This is what is left after the 'prime tail' A261130(n) and the 'prime head' A006519(m) = A001316(n) have been cut off.

Peter Luschny, Swing, divide and conquer the factorial, (excerpt).
Peter Luschny, Fast Factorial Functions, a code repository.
Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture.

Cf. A000265, A263931, A261130, A006519, A000984, A001316, A059097, A056040.

A263931 := n -> binomial(2*n, n) / convert(select(isprime, {$n+1..2*n}), `*`):
A000265 := n -> n / 2^padic[ordp](n, 2):
seq(A000265(A263931(n)), n = 0..45);

def A356637(n: int) -> int:
    m = 2 * n
    if m < 5: return 1
    sqrtm = isqrt(m) + 1
    R = prime_range(sqrtm, m // 3 + 1)
    factors = [x for x in R if is_odd(m // x)]
    for prime in prime_range(3, sqrtm):
        p: int = 1
        q: int = m
        while True:
            q //= prime
            if q == 0:
                break
            if q & 1 == 1:
                p *= prime
        if p > 1:
            factors.append(p)
    return product(factors)
print([A356637(n) for n in range(45)])

A000984(n) = a(n) * A001316(n) * A261130(n) for n >= 2.

A163774 Row sums of the central coefficients triangle (A163771).

Original entry on oeis.org

1, 3, 13, 51, 201, 783, 3039, 11763, 45481, 175803, 679779, 2630367, 10187659, 39500373, 153329913, 595883763, 2318471289, 9030982491, 35216266947, 137469149451, 537152523711, 2100857828193, 8223917499477, 32219655346719, 126328429601451, 495676719721953, 1946227355491909

Offset: 0

Peter Luschny, Aug 05 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A163771.

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:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i),i=k..n), k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$, where i$ denotes the swinging factorial of i (A056040).

Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - Werner Schulte, Nov 17 2015

More terms from Michel Marcus, Nov 24 2015

A181861 a(n) = gcd(n^2, n!/floor(n/2)!^2).

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 4, 7, 2, 9, 4, 11, 12, 13, 4, 45, 2, 17, 4, 19, 4, 21, 4, 23, 4, 25, 4, 27, 8, 29, 180, 31, 2, 99, 4, 175, 12, 37, 4, 117, 20, 41, 12, 43, 8, 675, 4, 47, 36, 49, 4, 153, 8, 53, 4, 55, 56, 57, 4, 59, 16

Offset: 0

Peter Luschny, Nov 21 2010

nonn

Marius A. Burtea, Table of n, a(n) for n = 0..1000

Cf. A170826, A181860, A056040, A000290.

[Gcd(n^2, Floor(Factorial(n)/(Factorial(Floor(n/2))^2))):n in [0..60]]; // Marius A. Burtea, Aug 03 2019

A181861 := n -> igcd(n^2,n!/iquo(n,2)!^2);

sf[n_] := n!/Quotient[n, 2]!^2; Table[GCD[n^2, sf[n]], {n, 0, 60}] (* Jean-François Alcover, Jun 28 2013 *)

a(n)=gcd(n!/(n\2)!^2,n^2) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = gcd(A000290(n), A056040(n)).

A190909 Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 6, 1, 15, 70, 168, 54, 30, 1, 21, 140, 504, 270, 330, 20, 1, 28, 252, 1260, 990, 1980, 260, 140, 1, 36, 420, 2772, 2970, 8580, 1820, 2100, 70, 1, 45, 660, 5544, 7722, 30030, 9100, 16800, 1190, 630

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded as a generalization of the triangle A063007.
A063007(n,k) = binomial(n+k, n-k)*(2*k)$;
T(n,k) = binomial(n+k, n-k)*(k)$.
Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A056040(n) which can be seen as extended central binomial numbers.

			[0]  1
[1]  1,  1
[2]  1,  3,   2
[3]  1,  6,  10,    6
[4]  1, 10,  30,   42,   6
[5]  1, 15,  70,  168,  54,   30
[6]  1, 21, 140,  504, 270,  330,  20
[7]  1, 28, 252, 1260, 990, 1980, 260, 140

Peter Luschny, The lost Catalan numbers
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Cf. Row sums: A190910; A056040, A063007, A085478, A088617.

A190909 := (n,k) -> binomial(n+k,n-k)*k!/iquo(k,2)!^2:
seq(print(seq(A190909(n,k),k=0..n)),n=0..7);

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 25 2012 *)

T(n,1) = A000217(n). T(n,2) = 2*binomial(n+2,4) (Cf. A034827).

A193282 a(n) = (n!/floor(n/2)!)^2.

Original entry on oeis.org

1, 1, 4, 36, 144, 3600, 14400, 705600, 2822400, 228614400, 914457600, 110649369600, 442597478400, 74798973849600, 299195895398400, 67319076464640000, 269276305858560000, 77820852393123840000, 311283409572495360000, 112373310855670824960000

Offset: 0

Peter Luschny, Sep 08 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000142, A056040, A195009.

[(Factorial(n)/Factorial(Floor(n/2)))^2: n in [0..20]]; // Vincenzo Librandi, Sep 11 2011

Maple
```
A193282 := n -> (n!/iquo(n,2)!)^2;
```

Table[(n!/(Floor[n/2]!))^2,{n,0,20}] (* Harvey P. Dale, Jul 30 2020 *)

a(n) = A056040(n)*A000142(n).
a(n) = A081125(n)^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A195009(n,k).
a(n) = n!^2*[x^n] (1+x)*BesselI(0,2*x). Here [x^n]f(x) denotes the coefficient of x^n in f(x).
Conjecture: a(n) + 8*a(n-1) - 4*(n-2)*(n+2)*a(n-2) + 16*(-2*n^2 + 6*n - 3)*a(n-3) - 64*(n-3)^2*a(n-4) = 0. - R. J. Mathar, Oct 03 2014

A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.

Original entry on oeis.org

1, 0, 2, 2, 8, 17, 49, 128, 356, 983, 2759, 7779, 22087, 63000, 180478, 518846, 1496236, 4326383, 12539335, 36419069, 105971473, 308866226, 901573732, 2635235789, 7712078755, 22594899002, 66266698424, 194531585078, 571561286576, 1680679630089, 4945738222801

Offset: 0

Peter Luschny, Aug 30 2011

nonn

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A194589.

A189912 := n -> add(n!/((n-k)!*iquo(k,2)!^2 *(iquo(k,2)+1)),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):

a[0] = 1; a[n_] := a[n] = Sum[(n-1)!/((n-k-1)!*Quotient[k, 2]!^2*(1 + Quotient[k, 2])), {k, 0, n-1}] - a[n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 30 2013 *)

a(n) = ((n+1) mod 2) + (1/2)*sum_{k=1..n}((-1)^k*binomial(n,k)*((k+1)/2)^(k mod 2)*(k+1)$+2*(-1)^n*(2*k)$/(k+1)), where n$ denotes the swinging factorial A056040(n).

cp's OEIS Frontend

A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

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A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

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A181860 a(n) = lcm(n^2, swinging_factorial(n)).

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A189911 Row sums of the extended Catalan triangle A189231.

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A356637 a(n) = A000265(A263931(n)).

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A163774 Row sums of the central coefficients triangle (A163771).

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A181861 a(n) = gcd(n^2, n!/floor(n/2)!^2).

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