A081124 -id:A081124 - cp's OEIS frontend

A081123 a(n) = floor(n/2)!.

1, 1, 1, 1, 2, 2, 6, 6, 24, 24, 120, 120, 720, 720, 5040, 5040, 40320, 40320, 362880, 362880, 3628800, 3628800, 39916800, 39916800, 479001600, 479001600, 6227020800, 6227020800, 87178291200, 87178291200, 1307674368000, 1307674368000, 20922789888000, 20922789888000

Offset: 0

Paul Barry, Mar 07 2003

This is the product of the first parts of the partitions (as nondecreasing list of parts) of n with exactly two positive integer parts, n > 1. - Wesley Ivan Hurt, Jan 25 2013

			a(8) = 24, since 8 has 4 nondecreasing partitions with exactly two positive integer parts: (1,7),(2,6),(3,5),(4,4).  Multiplying the first parts of these partitions together, we get: (1)(2)(3)(4) = 4! = 24. - _Wesley Ivan Hurt_, Jun 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000142, A004526, A081124, A081125.

[Factorial(Floor(n/2)): n in [0..40]]; // Vincenzo Librandi, Aug 06 2013

a:=n->floor(n/2)!: seq(a(k),k=1..70); # Wesley Ivan Hurt, Jun 03 2013

Table[(Floor[n/2])!, {n, 0, 40}] (* Vincenzo Librandi, Aug 06 2013 *)

for(n=0,50, print1((n\2)!, ", ")) \\ G. C. Greubel, Aug 01 2017

a(n) = floor(n/2)!.
E.g.f.: 1+sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2). - Vladeta Jovovic, Sep 25 2003
From Sergei N. Gladkovskii, Jul 28 2012: (Start)
G.f. G(0) where G(k) = 1 + x/(1 - x*(k+1)/( x*(k+1) + 1/G(k+1))); (continued fraction, 3rd kind, 3-step ).
E.g.f. 1 + sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2) = 1 + x/(G(0)-x) where G(k) =  2*k + 1 + x - (2*k+1)*x/(x + 2 - 2*x/G(k+1)); (continued fraction, 1st kind, 2-step).
(End)
G.f.: U(0) where U(k) = 1 + x/(1 - x*(k+2)/(x*(k+2) + 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: U(0) where U(k) =  1 + x/((2*k+1) - x*(2*k+1)/(x + 2*1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: 1 + x*G(0) where G(k) = 1 + x*(k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012

A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 34, 21, 6, 1, 1, 7, 31, 73, 73, 31, 7, 1, 1, 8, 43, 136, 209, 136, 43, 8, 1, 1, 9, 57, 229, 501, 501, 229, 57, 9, 1, 1, 10, 73, 358, 1045, 1546, 1045, 358, 73, 10, 1, 1, 11, 91, 529, 1961, 4051, 4051, 1961

Offset: 0

Michael Somos, Oct 08 2003

A(n,m) is the number of ways to pair the elements of two sets (with respectively n and m elements), where each element of either set may be paired with zero or one elements of the other set; number of n X m matrices of zeros and ones with at most one one in each row and column. E.g., A(2,2)=7 because we can pair {A,B} with {C,D} as {AB,CD}, {AC,BD}, {AC,B,D}, {AD,B,C}, {BC,A,D}, {BD,A,C}, or {A,B,C,D}. - Franklin T. Adams-Watters, Feb 06 2006
Compare with A086885. - Peter Bala, Sep 17 2008
A(n,m) is the number of vertex covers and independent vertex sets in the n X m lattice (rook) graph K_n X K_m. - Andrew Howroyd, May 14 2017

			      1       1       1       1       1       1       1       1       1
      1       2       3       4       5       6       7       8       9
      1       3       7      13      21      31      43      57      73
      1       4      13      34      73     136     229     358     529
      1       5      21      73     209     501    1045    1961    3393
      1       6      31     136     501    1546    4051    9276   19081
      1       7      43     229    1045    4051   13327   37633   93289
      1       8      57     358    1961    9276   37633  130922  394353
      1       9      73     529    3393   19081   93289  394353 1441729

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. J. Mathar, The number of binary nXm matrices with at most k 1's in each row or column, (2014) Table 1.
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Wikipedia, Rook polynomial

Row sums give A081124.
Main diagonal is A002720.
Cf. A099597, A176120.

A088699 := proc(i,j)
    add(binomial(i,k)*binomial(j,k)*k!,k=0..min(i,j)) ;
end proc: # R. J. Mathar, Feb 28 2015

max = 11; se = Series[E^x/(1 - y - x*y), {x, 0, max}, {y, 0, max}] // Normal // Expand; a[i_, j_] := SeriesCoefficient[se, {x, 0, i}, {y, 0, j}]*i!; Flatten[ Table[ a[i - j, j], {i, 0, max}, {j, 0, i}]] (* Jean-François Alcover, May 15 2012 *)

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x+x*O(x^i))*(1+x)^j,i))

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x/(1-x)+x*O(x^i))*(1-x)^(i-j-1),i))

A(i,j)=local(M); if(i<0 || j<0,0,M=matrix(j+1,j+1,n,m,if(n==m,1,if(n==m+1,m))); (M^i)[j+1,]*vectorv(j+1,n,1)) /* Michael Somos, Jul 03 2004 */

E.g.f.: exp(x)/(1-y-xy)=Sum_{i, j} A(i, j) y^j x^i/i!.
A(i, j) = A(i-1, j)+j*A(i-1, j-1)+(i==0) = A(j, i).
T(n, k) = sum{j=0..k, C(n, k-j)*k!/j!} = sum{j=0..k, (k-j)!*C(k, j)C(n, k-j)}. - Paul Barry, Nov 14 2005
A(i,j) = sum_k C(i,k)*C(j,k)*k!. E.g.f.: sum_{i,j} a(i,j)*x^i/i!*y^j/j! = e^{x+y+xy}. - Franklin T. Adams-Watters, Feb 06 2006
The LDU factorization of this array, formatted as a square array, is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!, 1!, 2!, ... ). Compare with A099597. - Peter Bala, Nov 06 2007
A(i,j) = (-1)^-i HypergeometricU(-i, 1 - i + j, -1). - Eric W. Weisstein, May 10 2017

A114162 C(n,k)*Floor((n-k)/2)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 4, 6, 4, 1, 2, 10, 10, 10, 5, 1, 6, 12, 30, 20, 15, 6, 1, 6, 42, 42, 70, 35, 21, 7, 1, 24, 48, 168, 112, 140, 56, 28, 8, 1, 24, 216, 216, 504, 252, 252, 84, 36, 9, 1, 120, 240, 1080, 720, 1260, 504, 420, 120, 45, 10, 1

Offset: 0

Paul Barry, Nov 14 2005

Row sums are A081124.

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  2, 4, 6, 4, 1;
  2, 10, 10, 10, 5, 1;
  6, 12, 30, 20, 15, 6, 1;
  6, 42, 42, 70, 35, 21, 7, 1;

Cf. A081124.

T(n, k) = if(k<=n, C(n, k)floor((n-k)/2)!, 0).

A361522 The aerated factorial numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 0, 24, 0, 120, 0, 720, 0, 5040, 0, 40320, 0, 362880, 0, 3628800, 0, 39916800, 0, 479001600, 0, 6227020800, 0, 87178291200, 0, 1307674368000, 0, 20922789888000, 0, 355687428096000, 0, 6402373705728000, 0, 121645100408832000, 0, 2432902008176640000

Offset: 0

Peter Luschny, Mar 14 2023

nonn

An aerated version of A000142, which is the main entry for this sequence.

Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Eric Weisstein's World of Mathematics, Error function erf.

Cf. A000142, A081124, A084261, A081123, A161556.

egf := (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1:
ser := series(egf, z, 42): seq(n!*coeff(ser, z, n), n = 0..40);

a[n_] := If[OddQ[n], 0, (n/2)!]; Array[a, 41, 0] (* Amiram Eldar, Mar 14 2023 *)

a(n) = n! * [z^n] (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1.

a(n) = n! * [z^n] 1 + 2*u*exp(u)*hypergeom([1/2], [3/2], -u), where u = (z/2)^2.

cp's OEIS Frontend

A081123 a(n) = floor(n/2)!.

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A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

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A114162 C(n,k)*Floor((n-k)/2)!.

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A361522 The aerated factorial numbers.

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