A056040 -id:A056040 - cp's OEIS frontend

A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

1, 5, 21, 715, -162877, 19840275, -7176079695, 1829885835675, -5009184735027165, 2216222559226679575, -2463196751104762933637, 1679951011110471133453965, -5519118103058048675551057049, 5373485053345792589762994345215, -12239617587594386225052760043303511

Offset: 0

Peter Luschny, Dec 27 2012

sign

Let N = 4*n+3 and A = sum_{k>=0} a(k)/(A123854(k)*N^(2*k)) then
C(n) ~ 8*4^n*A/(N*sqrt(N*Pi)), C(n) = (4^n/sqrt(Pi))*(Gamma(n+1/2)/ Gamma(n+2)) the Catalan numbers A000108.
The asymptotic expansion of the Catalan numbers considered here is based on the Taylor expansion of square root of the sine cardinal. This asymptotic series involves only even powers of N, making it more efficient than the asymptotic series based on Stirling's approximation to the central binomial which involves all powers (see for example: D. E. Knuth, 7.2.1.6 formula (16)). The series is discussed by Kessler and Schiff but is included as a special case in the asymptotic expansion given by J. L. Fields for quotients Gamma(x+a)/Gamma(x+b) and discussed by Y. L. Luke (p. 34-35), apparently overlooked by Kessler and Schiff.

			With N = 4*n+3 the first few terms of A are A = 1 + 5/(4*N^2) + 21/(32*N^4) + 715/(128*N^6) - 162877/(2048*N^8) + 19840275/(8192*N^10). With this A C(n) = round(8*4^n*A/(N*sqrt(N*Pi))) for n = 0..39 (if computed with sufficient numerical precision).

Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees—History of Combinatorial Generation, 2006.
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

Peter Luschny, Table of n, a(n) for n = 0..99
J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.

The logarithmic version is A220422. Appears in A193365 and A220466.

Cf. A220412.

A220002 := proc(n) local s; s := n -> `if`(n > 0, s(iquo(n,2))+n, 0);
(-1)^n*mul(4*i+2, i = 1..2*n)*2^s(iquo(n,2))*coeff(taylor(sqrt(sin(x)/x), x,2*n+2), x, 2*n) end: seq(A220002(n), n = 0..14);
# Second program illustrating J. L. Fields expansion of gamma quotients.
A220002 := proc(n) local recF, binSum, swing;
binSum := n -> add(i,i=convert(n,base,2));
swing := n -> n!/iquo(n, 2)!^2;
recF := proc(n, x) option remember; `if`(n=0, 1, -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*recF(n-2*k-2,x),k=0..n/2-1)) end: recF(2*n,-1/4)*2^(3*n-binSum(n))*swing(4*n+1) end:

max = 14; CoefficientList[ Series[ Sqrt[ Sinc[x]], {x, 0, 2*max+1}], x^2][[1 ;; max+1]]*Table[ (-1)^n*Product[ (2*k+1), {k, 1, 2*n}], {n, 0, max}] // Numerator (* Jean-François Alcover, Jun 26 2013 *)

length = 15; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
def A005187(n): return A005187(n//2) + n if n > 0 else 0
def A220002(n):
    P = mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
    return (-1)^n*P*T.coefficient(x, 2*n)
[A220002(n) for n in range(length)]

# Second program illustrating the connection with the Euler numbers.
def A220002_list(n):
    S = lambda n: sum((4-euler_number(2*k))/(4*k*x^(2*k)) for k in (1..n))
    T = taylor(exp(S(2*n+1)),x,infinity,2*n-1).coefficients()
    return [t[0].numerator() for t in T][::-1]
A220002_list(15)

Let [x^n]T(f(x)) denote the coefficient of x^n in the Taylor expansion of f(x) then r(n) = (-1)^n*prod_{i=1..2n}(2i+1)*[x^(2*n)]T(sqrt(sin(x)/x)) is the rational coefficient of the asymptotic expansion (in N=4*n+3) and a(n) = numerator(r(n)) = r(n)*2^(3*n-bs(n)), where bs(n) is the binary sum of n (A000120).
Also a(n) = numerator([x^(2*n)]T(exp(S))) where S = sum_{k>=1}((4-E(2*k))/ (4*k)*x^(2*k)) and E(n) the Euler numbers A122045.
Also a(n) = sf(4*n+1)*2^(3*n-bs(n))*F_{2*n}(-1/4) where sf(n) is the swinging factorial A056040, bs(n) the binary sum of n and F_{n}(x) J. L. Fields' generalized Bernoulli polynomials A220412.
In terms of sequences this means
r(n) = (-1)^n*A103639(n)*A008991(n)/A008992(n),
a(n) = (-1)^n*A220371(n)*A008991(n)/A008992(n).
Note that a(n) = r(n)*A123854(n) and A123854(n) = 2^A004134(n) = 8^n/2^A000120(n).
Formula from Johannes W. Meijer:
a(n) = d(n+1)*A098597(2*n+1)*(A008991(n)/A008992(n)) with d(1) = 1 and
d(n+1) = -4*(2*n+1)*A161151(n)*d(n),
d(n+1) = (-1)^n*2^(-1)*(2*(n+1))!*A060818(n)*A048896(n).

A107373 a(n) = (n/2)*binomial(n-1, floor((n-1)/2)) - 2^(n-2).

Original entry on oeis.org

0, 0, 1, 2, 7, 14, 38, 76, 187, 374, 874, 1748, 3958, 7916, 17548, 35096, 76627, 153254, 330818, 661636, 1415650, 2831300, 6015316, 12030632, 25413342, 50826684, 106853668, 213707336, 447472972, 894945944, 1867450648, 3734901296, 7770342787, 15540685574

Offset: 1

N. J. A. Sloane, Jul 20 2007

nonn

Total number of descents in all faro permutations of length n-1. Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one. See also A340567, A340568 and A340569. - Sergey Kirgizov, Jan 11 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.
F. Disanto and S. Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A. 22:1 (2011), 39-60.
Igor Pak, The area of cyclic polygons: Recent progress on Robbins' Conjectures, Adv. Applied Math. 34 (2005), 690-696. Special issue in memory of David Robbins.

Cf. A131019, A131020, A131021.

[(n/2)*Binomial(n-1, Floor((n-1)/2)) - 2^(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 01 2013

A056040 := n -> n!/iquo(n,2)!^2:
A133265 := n -> (n+2+(n-2)*(-1)^n)/2:
A107373 := n -> (A056040(n)*A133265(n)-2^n)/4:
seq(A107373(n),n=1..34); # Peter Luschny, Aug 30 2011

Table[(n/2) Binomial[n-1, Floor[(n-1)/2]] - 2^(n-2), {n, 1, 40}] (* Vincenzo Librandi, Oct 01 2013 *)

a(2*n) = 2*A000531(n-1); a(2*n+1) = A000531(n). - Max Alekseyev, Sep 30 2013

(1-n)*a(n) + 2*(n-1)*a(n-1) + 4*(n-2)*a(n-2) + 8*(-n+2)*a(n-3) = 0. - R. J. Mathar, May 26 2013

A137762 Central elements in writing first the numerator and then the denominator (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 5, 6, 31, 30, 209, 140, 1471, 630, 10625, 2772

Offset: 1

Mohammad K. Azarian, Feb 13 2008

			1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3; --> 5 6
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A003506, A046201, A046204, A046205, A046206, A046208, A046212, A137752, A137753, A137754, A137755, A137756, A137757, A137758, A137759, A137760, A137761.

From M. F. Hasler, Jan 25 2012: (Start)
a(2*n-1) = A137763(2*n).
a(2*n) = A137763(2*n-1) = A046212(2*n-1) = A056040(2*n-1) = A100071(2*n-1). (End)

A137763 Central elements in writing first the denominator and then the numerator(left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 6, 5, 30, 31, 140, 209, 630, 1471, 2772, 10625

Offset: 1

Mohammad K. Azarian, Feb 13 2008

			1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3;  --> 6 5
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A046201, A046204, A046205, A046206, A046208, A046212, A137752, A137753, A137754, A137755, A137756, A137757, A137758, A137759, A137760, A137761, A137762.

a(2n) = A137762(2n-1); a(2n-1) = A137762(2n) = A046212(2n-1) = A056040(2n-1) = A100071(2n-1). - M. F. Hasler, Jan 25 2012

A162246 Swinging polynomials, coefficients read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 6, 3, 3, 1, 1, 4, 4, 12, 6, 12, 4, 4, 1, 1, 5, 5, 20, 10, 30, 10, 20, 5, 5, 1, 1, 6, 6, 30, 15, 60, 20, 60, 15, 30, 6, 6, 1, 1, 7, 7, 42, 21, 105, 35, 140, 35, 105, 21, 42, 7, 7, 1

Offset: 0

Peter Luschny, Jun 28 2009

Let p(n,x) = (1+x^2)^n+n*x*(1+x^2)^(n-1), then T(n,k) are the coefficients of these polynomials, read by rows, n = 0,1,...
The central numbers of the rows, i.e., the coefficients of x^n of p(n,x), are the swinging factorial numbers A056040(n).
Row sums: sum_{k=0..2n} T(n,k) = A001792(n).
sum_{k=0..2n} isodd(n+k)T(n,k) = 2^n(isodd(n)+(n/2)isodd(n+1))
= 0, 2, 4, 8, 32, 32, 192, 128, 1024, 512, 5120, ...
sum_{k=0..2n} iseven(n+k)T(n,k) = 2^n(isodd(n)(n/2)+isodd(n+1))
= 1, 1, 4, 12, 16, 80, 64, 448, 256, 2304, 1024, ...

			The central coefficients are marked by [].
[1]
1,[1],1
1,2,[2],2,1
1,3,3,[6],3,3,1
1,4,4,12,[6],12,4,4,1
1,5,5,20,10,[30],10,20,5,5,1
1,6,6,30,15,60,[20],60,15,30,6,6,1
1,7,7,42,21,105,35,[140],35,105,21,42,7,7,1
p(0,x) = 1
p(1,x) = x^2+x+1
p(2,x) = x^4+2x^3+2x^2+2x+1
p(3,x) = x^6+3x^5+3x^4+6x^3+3x^2+3x+1
p(4,x) = x^8+4x^7+4x^6+12x^5+6x^4+12x^3+4x^2+4x+1
p(5,x) = x^10+5x^9+5x^8+20x^7+10x^6+30x^5+10x^4+20x^3+5x^2+5x+1

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A056040, A001792.

p := (n,x) -> (1+x^2)^n+n*x*(1+x^2)^(n-1):
seq(print(seq(coeff(expand(p(n,x)),x,i),i=0..2*n)),n=0..7);
T := (n,k) -> n!/((n-ceil(k/2))!*floor(k/2)!);
seq(print(seq(T(n,k),k=0..2*n)),n=0..7);

t[n_, k_] := If[EvenQ[k], Binomial[n, k/2], Binomial[n, (k-1)/2]*(n-(k-1)/2)]; Table[t[n, k], {n, 0, 7}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = n!/((n-ceiling(k/2))!*floor(k/2)!).

A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

Original entry on oeis.org

1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675, 439793834236745, 2251006269442815, 11509340056410735, 58790764269668805

Offset: 0

Peter Luschny, Aug 06 2009

nonn

Also a(n) = Sum_{i=0..n} 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. A163842, A163872, A387208.

a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:

CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(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[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 26 2013 *)
Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
(End)
a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - Vladimir Reshetnikov, Apr 25 2016
E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - Ilya Gutkovskiy, Nov 19 2021
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)

A163209 Catalan pseudoprimes: odd composite integers n=2m+1 satisfying A000108(m) == (-1)^m 2 (mod n).

Original entry on oeis.org

5907, 1194649, 12327121

Offset: 1

Peter Luschny, Jul 24 2009

Also, Wilson spoilers: composite n which divide A056040(n-1) - (-1)^floor(n/2). For the factorial function, a Wilson spoiler is a composite n that divides (n-1)! + (-1). Lagrange proved that no such n exists. For the swinging factorial (A056040), the situation is different.
Also, composite odd integers n=2*m+1 such that A000984(m) == (-1)^m (mod n).
Contains squares of A001220. In particular, a(2) = A001220(1)^2 = 1093^2 = 1194649 = A001567(274) and a(3) = A001220(2)^2 = 3511^2 = 12327121 = A001567(824).
See the Vardi reference for a binomial setup.
Aebi and Cairns 2008, page 9: a(4) either has more than 2 factors or is > 10^10. - Dana Jacobsen, May 27 2015
a(4) > 10^10. - Dana Jacobsen, Mar 03 2018

I. Vardi, Computational Recreations in Mathematica, 1991, p. 66.

C. Aebi, G. Cairns (2008). "Catalan numbers, primes and twin primes". Elemente der Mathematik 63 (4): 153-164. doi:10.4171/EM/103
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Index entries for sequences related to pseudoprimes

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:
WS := proc(f,r,n) select(p->(f(p-1)+r(p)) mod p = 0,[$2..n]);
select(q -> not isprime(q),%) end:
A163209 := n -> WS(swing,p->(-1)^iquo(p+2,2),n);

v(n,p)=my(s); n*=2; while(n\=p, s+=n%2); s
is(n)=if(n%2==0,return(0)); my(m=Mod(1,n),a=n\2); fordiv(n,d,if(isprime(d) && v(a,d), return(0))); forprime(p=2,a, m*=p^v(a,p)); forprime(p=a+1,n,m*=p); m==(-1)^a
forcomposite(n=4,2e7, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, Mar 06 2015

# Reasonable for isolated values, slow for the sequence:
use ntheory ":all";
sub is { my $m = ($[0]-1)>>1; (binomial($m<<1,$m) % $[0]) == (($m&1) ? $_[0]-1 : 1); }
foroddcomposites { say if is($) } 2e7;  # _Dana Jacobsen, May 03 2015

# Much faster for sequential testing:
use Math::GMPz; use ntheory ":all"; { my($c,$l)=(Math::GMPz->new(1),1); sub catalan { while ($[0] > $l) { $l++; $c *= 4*$l-2; Math::GMPz::Rmpz_divexact_ui($c,$c,$l+1); } $c; } } my $m; foroddcomposites { $m = ($-1)>>1; say if (catalan($m) % $) == (($m&1) ? $-2 : 2); } 2e7;  # Dana Jacobsen, May 03 2015

a(1) = 5907 = 3*11*179 was found by S. Skiena
Typo corrected Peter Luschny, Jul 25 2009
Edited by Max Alekseyev, Jun 22 2012

A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

Original entry on oeis.org

0, 1, 2, 3, 12, 10, 60, 35, 280, 126, 1260, 462, 5544, 1716, 24024, 6435, 102960, 24310, 437580, 92378, 1847560, 352716, 7759752, 1352078, 32449872, 5200300, 135207800, 20058300, 561632400, 77558760, 2326762800, 300540195, 9617286240, 1166803110, 39671305740

Offset: 0

Peter Luschny, Oct 24 2013

nonn

a(n) + A056040(n) = A189911(n), the row sums of the extended Catalan triangle A189231.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.

Cf. A005430, A001700, A056040, A189911.

A212303 := proc(n) if n mod 2 = 0 then n*binomial(n, iquo(n,2))/2 else binomial(n+1, iquo(n,2)+1)/2 fi end: seq(A212303(i), i=0..36);

a[n_?EvenQ] := n*Binomial[n, n/2]/2; a[n_?OddQ] := Binomial[n+1, Quotient[n, 2]+1]/2; Table[a[n], {n, 0, 36}]  (* Jean-François Alcover, Feb 05 2014 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a(n+1),(4a(n+1))/(n(n+2))]}; Join[{0}, Transpose[ NestList[ nxt,{1,1},40]][[2]]] (* Harvey P. Dale, Dec 20 2014 *)

def A212303():
    yield 0
    r, n = 1, 1
    while True:
        yield r
        n += 1
        r *= n if is_even(n) else 4*n/((n-1)*(n+1))
a = A212303(); [next(a) for i in range(36)]

E.g.f.: (1+x)*BesselI(1, 2*x).
O.g.f.: -((4*x^2-1)^(3/2)+I-(4*I)*x^2+(4*I)*x^3)/(2*x*(4*x^2-1)^(3/2)).
Recurrence: a(n) = n if n < 2 else a(n) = a(n-1)*n if n is even else a(n-1)*n*4/((n-1)*(n+1)).
a(2*n) = n*C(2*n, n) (A005430); a(2*n+1) = C(2*n+1,  n+1) (A001700).
a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.
a(n) = Sum_{k=0..n} A189231(n, 2*k+1).
Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 4, 5, 6, 5, 4, 2, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 2, 4, 7, 12, 17, 24, 31, 39, 45, 51, 54, 56, 54, 51, 45, 39, 31, 24, 17, 12, 7, 4, 2, 1

Offset: 0

Peter Luschny, Jul 19 2016

The q-swing_factorial(n) is a univariate polynomial over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2) and at least floor(n/2) irreducible factors.
Evaluated at q=1 q-swing_factorial(n) gives the swinging factorial A056040(n).
Combinatorial interpretation: The definition of an orbital system is given in A232500 and in the link 'Orbitals'. The number of orbitals over n sectors is counted by the swinging factorial.
The major index of an orbital is the sum of the positions of steps which are immediately followed by a step with strictly smaller value. This statistic is an extension of the major index statistic given in A063746 which reappears in the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. As in the case of the central binomial also in the case of the swinging factorial the major index coincides with its q-analog.

			The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q + 1)
[5] (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q + 1) * (q^2 - q + 1) * (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
The coefficients of the polynomials start:
[n] [k=0,1,2,...] [row sum]
[0] [1] [1]
[1] [1] [1]
[2] [1, 1] [2]
[3] [1, 2, 2, 1] [6]
[4] [1, 1, 2, 1, 1] [6]
[5] [1, 2, 4, 5, 6, 5, 4, 2, 1] [30]
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] [20]
[7] [1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1] [140]
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] [70]
T(5, 4) = 6 because the 2 orbitals [-1,-1,1,1,0] and [-1,0,1,1,-1] have at position 4 and the 4 orbitals [0,-1,1,-1,1], [1,-1,0,-1,1], [1,-1,1,-1,0] and [1,0,1,-1,-1] at positions 1 and 3 a down step.

Peter Luschny, Orbitals

Cf. A056040 (row sums), A274887 (q-factorial), A063746 (q-central_binomial).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

QSwingFactorial_coeffs := proc(n) local P,a,b;
a := mul((p^(n-i)-1)/(p^(i+1)-1),i=0..iquo(n,2)-1);
b := ((p^(iquo(n,2)+1)-1)/(p-1))^((1-(-1)^n)/2);
P := simplify(a*b); seq(coeff(P,p,j),j=0..degree(P)) end:
for n from 0 to 9 do print(QSwingFactorial_coeffs(n)) od;
# Alternatively (recursive):
with(QDifferenceEquations):
QSwingRec := proc(n,q) local r; if n = 0 then return 1 fi:
if irem(n,2) = 0 then r := (1+q^(n/2))/QBrackets(n/2,q)
else r := QBrackets(n,q) fi; r*QSwingRec(n-1,q) end:
Trow := proc(n) expand(QSimplify(QSwingRec(n,q)));
seq(coeff(%,q,j),j=0..degree(%)) end: seq(Trow(n),n=0..10);

p[n_] := QFactorial[n, q] / QFactorial[Quotient[n, 2], q]^2
Table[CoefficientList[p[n] // FunctionExpand, q], {n,0,9}] // Flatten

from sage.combinat.q_analogues import q_factorial
def q_swing_factorial(n, q=None):
    return q_factorial(n)//q_factorial(n//2)^2
for n in (0..8): print(q_swing_factorial(n).list())

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_major_index(n):
    S = [0]*(((n+1)//2)^2 + ((n+1) % 2))
    for u in unit_orbitals(n):
        L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]
        # i+1 because u is 0-based whereas convention assumes 1-base.
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_major_index(n))

q_swing_factorial(n) = q_factorial(n)/q_factorial(floor(n/2))^2.
q_swing_factorial(n) = q_binomial(n-eta(n),floor((n-eta(n))/2))*q_int(n)^eta(n) with eta(n) = (1-(-1)^n)/2.
Recurrence: q_swing_factorial(0,q) = 1 and for n>0 q_swing_factorial(n,q) = r*q_swing_factorial(n-1,q) with r = (1+q^(n/2))/[n/2;q] if n is even else r = [n;q]. Here [a;q] are the q_brackets.
The generating polynomial for row n is P_n(p) = ((p^(floor(n/2)+1)-1)/(p-1))^((1-(-1)^n)/2)*Product_{i=0..floor(n/2)-1}((p^(n-i)-1)/(p^(i+1)-1)).

A327493 a(n) = 2^A327492(n).

Original entry on oeis.org

1, 4, 8, 32, 128, 512, 1024, 4096, 32768, 131072, 262144, 1048576, 4194304, 16777216, 33554432, 134217728, 2147483648, 8589934592, 17179869184, 68719476736, 274877906944, 1099511627776, 2199023255552, 8796093022208, 70368744177664, 281474976710656, 562949953421312

Offset: 0

Peter Luschny, Sep 27 2019

nonn

Cf. A327492, A327491, A327494, A327495, A056040.

bitcount(n) = sum(digits(n, base = 2))
A327493(n) = 2^(2n - bitcount(n) + mod(n, 2))
[A327493(n) for n in 0:26] |> println # Peter Luschny, Oct 03 2019

A327493 := n -> 2^(A327492_list(n+1)[n+1]):
seq(A327493(n), n = 0..26);

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[n+1] = a[n] * 2^if(n%4, n%2 + 1, valuation(n,2))); a} \\ Andrew Howroyd, Sep 28 2019

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

a(n) = denominator(b(n)) where b(n) = n!/(2^n*floor(n/2)!)^2 is the normalized swinging factorial (A056040).

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