A065140 -id:A065140 - cp's OEIS frontend

A268363 Number of n X 2 arrays containing 2 copies of 0..n-1 with row sums equal.

1, 1, 4, 12, 96, 480, 5760, 40320, 645120, 5806080, 116121600, 1277337600, 30656102400, 398529331200, 11158821273600, 167382319104000, 5356234211328000, 91055981592576000, 3278015337332736000, 62282291409321984000, 2491291656372879360000

Offset: 0

R. H. Hardin, Feb 03 2016

nonn

Column 2 of A268367.

			Some solutions for n=5
..3..1....1..3....4..0....0..4....0..4....4..0....4..0....4..0....2..2....1..3
..1..3....2..2....3..1....3..1....2..2....3..1....1..3....2..2....0..4....0..4
..0..4....0..4....0..4....2..2....1..3....1..3....0..4....1..3....3..1....4..0
..0..4....0..4....2..2....1..3....4..0....0..4....2..2....0..4....4..0....3..1
..2..2....3..1....1..3....0..4....1..3....2..2....1..3....3..1....3..1....2..2

Alois P. Heinz, Table of n, a(n) for n = 0..425 (terms n = 1..25 from R. H. Hardin)

Cf. A065140, A121627, A268367.

Cf. A158867.

[2^Floor(n/2)*Factorial(n): n in [0..25]]; // G. C. Greubel, Mar 08 2022

Table[2^Floor[n/2] n!, {n,0,25}] (* Michael De Vlieger, Aug 11 2016 *)

[2^(n//2)*factorial(n) for n in (0..25)] # G. C. Greubel, Mar 08 2022

a(n) = 2^floor(n/2) * n!. - Joel B. Lewis, Aug 11 2016
a(2n) = A065140(n) for terms > a(1) - Terry D. Grant, May 28 2017
a(n) = A158867(n, n) for n > 0. - G. C. Greubel, Mar 08 2022

Title clarified by Joel B. Lewis, Aug 11 2016

a(0)=1 prepended by Alois P. Heinz, May 27 2017

A152029 a(n) = 2^n(2n)!/((n+1)!).

Original entry on oeis.org

1, 2, 16, 240, 5376, 161280, 6082560, 276756480, 14760345600, 903333150720, 62412108595200, 4805732361830400, 408117579035443200, 37896632339005440000, 3819980539771748352000, 415422883700177633280000, 48482294191832495554560000, 6044126009248451112468480000

Offset: 0

N. J. A. Sloane, Sep 15 2009

nonn

Robert Israel, Table of n, a(n) for n = 0..334
K. Casteels and B. Stevens, Universal cycles of (n-1)-partitions of an n-set, Discr. Math., 309 (2009), 5332-5340.

[2^n*Factorial(2*n)/Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Jan 27 2017

seq(2^n*(2*n)!/(n+1)!,n=0..40); # Robert Israel, Jan 25 2017

Table[(2^n) (2 n)! / (n + 1)!, {n, 0, 20}] (* Vincenzo Librandi, Jan 27 2017 *)
With[{nn=20},CoefficientList[Series[2/(1+(1-8x)^(1/2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 11 2023 *)

a(n) = 2^n*(2*n)!/(n+1)! \\ Michel Marcus, Jun 02 2013

E.g.f 2/(1+(1-8*x)^(1/2)). - Sergei N. Gladkovskii, Oct 26 2012
a(n) = 2^n * A001761(n) = A065140(n)/(n+1)!. - Michel Marcus, Jun 02 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + (k+2)/((2*k+1)*(2*k+2))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
4*(n+1)*(2*n+1)*a(n) = (n+2)*a(n+1). - Robert Israel, Jan 25 2017
E.g.f.: 1/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

A232205 a(0)=1; thereafter a(n) = na(n-1) if n is even, otherwise a(n) = 2n*a(n-1).

Original entry on oeis.org

1, 2, 4, 24, 96, 960, 5760, 80640, 645120, 11612160, 116121600, 2554675200, 30656102400, 797058662400, 11158821273600, 334764638208000, 5356234211328000, 182111963185152000, 3278015337332736000, 124564582818643968000, 2491291656372879360000, 104634249567660933120000, 2301953490488540528640000

Offset: 0

N. J. A. Sloane, Nov 21 2013

nonn

Bisections give: A065140 (even part), A122551 (odd part).

c:=proc(n) option remember; if n=0 then 1
elif (n mod 2) = 0 then n*c(n-1) else 2*n*c(n-1); fi; end;
[seq(c(n),n=0..20)];

nxt[{n_,a_}]:={n+1,If[OddQ[n],a(n+1),2a(n+1)]}; NestList[nxt,{0,1},30][[All,2]] (* Harvey P. Dale, Jul 20 2020 *)

a(n) = n!*2^floor((n+1)/2). - Jon E. Schoenfield, Nov 24 2013

Definition corrected to match terms by Jon E. Schoenfield, Nov 24 2013

cp's OEIS Frontend

A268363 Number of n X 2 arrays containing 2 copies of 0..n-1 with row sums equal.

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A152029 a(n) = 2^n(2n)!/((n+1)!).

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A232205 a(0)=1; thereafter a(n) = na(n-1) if n is even, otherwise a(n) = 2n*a(n-1).

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cp's OEIS Frontend

A268363 Number of n X 2 arrays containing 2 copies of 0..n-1 with row sums equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A152029 a(n) = 2^n*(2*n)!/((n+1)!).

Views

Author

Keywords

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A232205 a(0)=1; thereafter a(n) = n*a(n-1) if n is even, otherwise a(n) = 2*n*a(n-1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A152029 a(n) = 2^n(2n)!/((n+1)!).

A232205 a(0)=1; thereafter a(n) = na(n-1) if n is even, otherwise a(n) = 2n*a(n-1).