A162246 -id:A162246 - cp's OEIS frontend

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220

Offset: 0

Labos Elemer, Jul 25 2000

nonn

a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]

			a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Didier Guillet, On swinging factorials and the lonely runner conjecture (Text in French).
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

[(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011

SeriesCoeff := proc(s,n) series(s(w,n),w,n+2);
convert(%,polynom); coeff(%,w,n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)),k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0,1,n^irem(n,2)*(4/n)^irem(n+1,2)*a2(n-1)) end;
a3 := n -> n!/iquo(n,2)!^2;
g4 := z -> BesselI(0,2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4,n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5,n);
g6 := (z,n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6,n);
a7 := n -> combinat[multinomial](n,floor(n/2),n mod 2,floor(n/2));
h := n -> binomial(n,floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1),h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)

a(n)=n!/(n\2)!^2 \\ Charles R Greathouse IV, May 02 2011

def A056040():
    r, n = 1, 0
    while True:
        yield r
        n += 1
        r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)]  # Peter Luschny, Oct 24 2013

a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2n+1) = A046212(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022

Extended and edited by Peter Luschny, Jun 28 2009

A189231 Extended Catalan triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1

Offset: 0

Peter Luschny, May 01 2011

Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.
Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.
The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.
T(n,0) are the extended Catalan numbers A057977(n).

			The Laurent polynomials:
C(0,x) =                 0
C(1,x) =               x - 1/x
C(2,x) =         x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0]   1,
[1]   1,  1,
[2]   1,  2,  1,
[3]   3,  2,  3,  1,
[4]   2,  8,  3,  4,  1,
[5]  10,  5, 15,  4,  5,  1,
[6]   5, 30,  9, 24,  5,  6,  1,
[7]  35, 14, 63, 14, 35,  6,  7, 1,
    [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189230.

A189231_poly := (n,x)-> `if`(n=0,0,(x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n],seq(coeff(expand(A189231_poly(n,x)),x,k),k=1..n)),n=1..9);
A189231 := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1,k-1)+modp(n-k,2)*A189231(n-1,k)+A189231(n-1,k+1))) end:
seq(print(seq(A189231(n,k),k=0..n)),n=0..9);

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2]*t[n-1, k] + t[n-1, k+1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).
S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).
Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.

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

A189230 Complementary Catalan triangle read by rows.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 0, 8, 0, 4, 0, 10, 0, 15, 0, 5, 0, 0, 30, 0, 24, 0, 6, 0, 35, 0, 63, 0, 35, 0, 7, 0, 0, 112, 0, 112, 0, 48, 0, 8, 0, 126, 0, 252, 0, 180, 0, 63, 0, 9, 0, 0, 420, 0, 480, 0, 270, 0, 80, 0, 10, 0, 462, 0, 990, 0, 825, 0, 385, 0, 99, 0, 11, 0

Offset: 0

Peter Luschny, May 01 2011

T(n,k) = A189231(n,k)*((n - k) mod 2). For comparison: the classical Catalan triangle is A053121(n,k) = A189231(n,k)*((n-k+1) mod 2).

T(n,0) = A138364(n). Row sums: A100071.

			[0]  0,
[1]  1,  0,
[2]  0,  2,  0,
[3]  3,  0,  3,  0,
[4]  0,  8,  0,  4,  0,
[5] 10,  0, 15,  0,  5, 0,
[6]  0, 30,  0, 24,  0, 6, 0,
[7] 35,  0, 63,  0, 35, 0, 7, 0,
   [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189231.

A189230 := (n,k) -> A189231(n,k)*modp(n-k,2):
seq(print(seq(A189230(n,k),k=0..n)),n=0..11);

t[n_, k_] /; (k>n || k<0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2] t[n-1, k] + t[n-1, k+1];
T[n_, k_] := t[n, k] Mod[n-k, 2];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 24 2019 *)

A177375 Triangle t(n,k): the coefficient [x^k] of the series (1+x)^n + 2nx*(1+x)^(n-2), in row n, column k.

Original entry on oeis.org

1, 1, 3, 1, 6, 1, 1, 9, 9, 1, 1, 12, 22, 12, 1, 1, 15, 40, 40, 15, 1, 1, 18, 63, 92, 63, 18, 1, 1, 21, 91, 175, 175, 91, 21, 1, 1, 24, 124, 296, 390, 296, 124, 24, 1, 1, 27, 162, 462, 756, 756, 462, 162, 27, 1, 1, 30, 205, 680, 1330, 1652, 1330, 680, 205, 30, 1

Offset: 0

Roger L. Bagula, May 07 2010

Row sums are in A001792.

			1;
1, 3;
1, 6, 1;
1, 9, 9, 1;
1, 12, 22, 12, 1;
1, 15, 40, 40, 15, 1;
1, 18, 63, 92, 63, 18, 1;
1, 21, 91, 175, 175, 91, 21, 1;
1, 24, 124, 296, 390, 296, 124, 24, 1;
1, 27, 162, 462, 756, 756, 462, 162, 27, 1;
1, 30, 205, 680, 1330, 1652, 1330, 680, 205, 30, 1;

Cf. A162246

A177375 := proc(n,k)
    (1+x)^n+2*n*x*(1+x)^(n-2) ;
    coeftayl(%,x=0,k)
end proc: # R. J. Mathar, May 19 2013

p[x, 0, q_] := 1; p[x, 1, q_] := x + 1;
p[x_, n_, q_] := p[x, n, q] = (1 + x)^n + 2*q*n*x*(1 + x)^(n - 2);
Table[Flatten[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}]], {q, 1, 10}]

A237884 a(n) = (n!m)/(m!(m+1)!) where m = floor(n/2).

Original entry on oeis.org

0, 0, 1, 3, 4, 20, 15, 105, 56, 504, 210, 2310, 792, 10296, 3003, 45045, 11440, 194480, 43758, 831402, 167960, 3527160, 646646, 14872858, 2496144, 62403600, 9657700, 260757900, 37442160, 1085822640, 145422675, 4508102925, 565722720, 18668849760, 2203961430

Offset: 0

Peter Luschny, Feb 14 2014

nonn

A237884 := proc(n) m := iquo(n,2); (n!*m)/(m!*(m+1)!) end;
seq(A237884(n), n = 0..34);

CoefficientList[Series[-((-1 + Sqrt[1 - 4 x^2] -x (-1 + Sqrt[1 - 4 x^2] +
2 x (-3 + 2 Sqrt[1 - 4 x^2] +x (3 + 4 x - 2 Sqrt[1 - 4 x^2]))))/
(2 x^2 (1 - 4 x^2)^(3/2))), {x, 0, 30}], x] (* Benedict W. J. Irwin, Aug 15 2016 *)
Table[(n! #)/(#! (# + 1)!) &@ Floor[n/2], {n, 0, 34}] (* Michael De Vlieger, Aug 15 2016 *)

def A237884():
    r, s, n = 1, 0, 0
    while True:
        yield s
        n += 1
        r *= 4/n if is_even(n) else n
        s = r * (n//2)/(n//2+1)
a = A237884(); [next(a) for i in range(35)]

a(2*n) = A001791(n).
a(2*n+1) = A000917(n-1).
a(n) = n^(n mod 2)*binomial(2*floor(n/2), floor(n/2)-1).
a(n) = A162246(n, n+2) = n!/((n-ceiling((n+2)/2))!*floor((n+2)/2)!) if n > 1, otherwise 0.
a(n) = A056040(n)*floor(n/2)/(floor(n/2)+1).
a(n) + A056040(n) = A057977(n).
G.f.: -((p - 1 - x*(p - 1 + 2*x*(2*p - 3 + x*(3 + 4*x - 2*p))))/(2*x^2*p^3)), where p=sqrt(1-4*x^2). - Benedict W. J. Irwin, Aug 15 2016

A177276 Triangle T(n,k) with the coefficient [x^k] of the polynomial (1+x^2)^n + 2nx*(1+x^2)^(n-1) in row n, column k, 0<=k<=2n.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 2, 4, 1, 1, 6, 3, 12, 3, 6, 1, 1, 8, 4, 24, 6, 24, 4, 8, 1, 1, 10, 5, 40, 10, 60, 10, 40, 5, 10, 1, 1, 12, 6, 60, 15, 120, 20, 120, 15, 60, 6, 12, 1, 1, 14, 7, 84, 21, 210, 35, 280, 35, 210, 21, 84, 7, 14, 1, 1, 16, 8, 112, 28, 336, 56, 560, 70, 560, 56, 336, 28, 112

Offset: 0

Roger L. Bagula, May 06 2010

Row sums are 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264,..., A001787.

This is a generalization of A162246 to polynomials (1+x^2)^n + q*n*x*(1+x^2)^(n-1), here q=2.

			1;
1, 2, 1;
1, 4, 2, 4, 1;
1, 6, 3, 12, 3, 6, 1;
1, 8, 4, 24, 6, 24, 4, 8, 1;
1, 10, 5, 40, 10, 60, 10, 40, 5, 10, 1;
1, 12, 6, 60, 15, 120, 20, 120, 15, 60, 6, 12, 1;
1, 14, 7, 84, 21, 210, 35, 280, 35, 210, 21, 84, 7, 14, 1;
1, 16, 8, 112, 28, 336, 56, 560, 70, 560, 56, 336, 28, 112, 8, 16, 1;
1, 18, 9, 144, 36, 504, 84, 1008, 126, 1260, 126, 1008, 84, 504, 36, 144, 9, 18, 1;
1, 20, 10, 180, 45, 720, 120, 1680, 210, 2520, 252, 2520, 210, 1680, 120, 720, 45, 180, 10, 20, 1;

Cf. A162246

p[x_, n_] = (1 + x^2)^n + 2*n*x*(1 + x^2)^(n - 1);
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

cp's OEIS Frontend

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

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A189231 Extended Catalan triangle read by rows.

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A189911 Row sums of the extended Catalan triangle A189231.

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A189230 Complementary Catalan triangle read by rows.

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A177375 Triangle t(n,k): the coefficient [x^k] of the series (1+x)^n + 2*n*x*(1+x)^(n-2), in row n, column k.

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A237884 a(n) = (n!*m)/(m!*(m+1)!) where m = floor(n/2).

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A177276 Triangle T(n,k) with the coefficient [x^k] of the polynomial (1+x^2)^n + 2*n*x*(1+x^2)^(n-1) in row n, column k, 0<=k<=2n.

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A177375 Triangle t(n,k): the coefficient [x^k] of the series (1+x)^n + 2nx*(1+x)^(n-2), in row n, column k.

A237884 a(n) = (n!m)/(m!(m+1)!) where m = floor(n/2).

A177276 Triangle T(n,k) with the coefficient [x^k] of the polynomial (1+x^2)^n + 2nx*(1+x^2)^(n-1) in row n, column k, 0<=k<=2n.