A081123 -id:A081123 - cp's OEIS frontend

A081125 a(n) = n! / floor(n/2)!.

1, 1, 2, 6, 12, 60, 120, 840, 1680, 15120, 30240, 332640, 665280, 8648640, 17297280, 259459200, 518918400, 8821612800, 17643225600, 335221286400, 670442572800, 14079294028800, 28158588057600, 647647525324800, 1295295050649600

Offset: 0

Paul Barry, Mar 07 2003

Product of the largest parts in the partitions of n+1 into exactly two parts, n > 0. - Wesley Ivan Hurt, Jan 26 2013 (Clarified on Apr 20 2016)

			a(3) = 6 since 3+1 = 4 has two partitions into two parts, (3,1) and (2,2), and the product of the largest parts is 6. - _Wesley Ivan Hurt_, Jan 26 2013 (Clarified on Apr 20 2016)

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Index to divisibility sequences.

Cf. A004526, A056040, A081123, A000407 (bisection), A001813 (bisection).

[Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // Vincenzo Librandi, Sep 13 2011

Method 1)  a:=n->n!/floor(n/2)!; seq(a(k),k=0..40); # Wesley Ivan Hurt, Jun 03 2013
Method 2)  with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # Wesley Ivan Hurt, Jun 06 2013

Table[n!/Floor[n/2]!, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 20 2016 *)

a(n)=n!/(n\2)! \\ Charles R Greathouse IV, Sep 13 2011

from sympy import rf
def A081125(n): return rf((m:=n+1>>1)+(n+1&1),m) # Chai Wah Wu, Jul 22 2022

def a(n): return rising_factorial(ceil(n/2),floor(n/2))
[a(n) for n in range(26)]  # Peter Luschny, Oct 09 2013

E.g.f.: (1+x)*exp(x^2). - Vladeta Jovovic, Sep 24 2003
From Peter Luschny, Aug 07 2009: (Start)
a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial (A056040).
a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
E.g.f.: E(0) where E(k) = 1 + x/(1 - x/(x + (k+1)/E(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: G(0) where G(k) =  1 + x*(2*k+1)/(1 - 2*x/(2*x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
D-finite with recurrence a(n) +2*a(n-1) -2*n*a(n-2) +4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
From Wesley Ivan Hurt, Jun 06 2013: (Start)
a(n) = n!/(n-floor((n+1)/2))!.
a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial (A056040). - Peter Luschny, Oct 28 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)

A332558 a(n) is the smallest k such that n(n+1)(n+2)...(n+k) is divisible by n+k+1.

Original entry on oeis.org

4, 3, 2, 3, 4, 5, 4, 3, 5, 4, 6, 5, 6, 5, 4, 7, 6, 5, 4, 3, 6, 7, 6, 5, 4, 8, 7, 6, 6, 5, 8, 7, 6, 5, 4, 8, 7, 6, 5, 7, 6, 5, 10, 9, 8, 9, 8, 7, 6, 9, 8, 7, 6, 5, 4, 6, 12, 11, 10, 9, 8, 7, 6, 7, 6, 5, 12, 11, 10, 9, 8, 7, 6, 5, 8, 7, 6, 11, 10, 9, 8, 7, 6, 5

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

This is a multiplicative analog of A332542.

a(n) always exists because one can take k to be 2^m - 1 for m large.

Robert Israel, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004.14000 [math.NT], April 2020.

Cf. A061836 (k+1), A332559 (n+k+1), A332560 (the final product), A332561 (the quotient).
For records, see A333532 and A333533 (and A333537), which give the records in the essentially identical sequence A061836.
Additive version: A332542, A332543, A332544, A081123.
"Concatenate in base 10" version: A332580, A332584, A332585.

f:= proc(n) local k,p;
  p:= n;
  for k from 1 do
    p:= p*(n+k);
    if (p/(n+k+1))::integer then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 04 2020, after Maple *)

a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } \\ Jinyuan Wang, Feb 25 2020

PARI
```
\\ See Corneth link
```

def a(n):
    k, p = 1, n*(n+1)
    while p%(n+k+1): k += 1; p *= (n+k)
    return k
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 06 2021

a(n) = A061836(n) - 1 for n >= 1.

a(n + 1) >= a(n) - 1. a(n + 1) = a(n) - 1 mostly. - David A. Corneth, Apr 14 2020

A067994 Hermite numbers.

Original entry on oeis.org

1, 0, -2, 0, 12, 0, -120, 0, 1680, 0, -30240, 0, 665280, 0, -17297280, 0, 518918400, 0, -17643225600, 0, 670442572800, 0, -28158588057600, 0, 1295295050649600, 0, -64764752532480000, 0, 3497296636753920000, 0, -202843204931727360000, 0

Offset: 0

Eric W. Weisstein, Feb 07 2002

sign

|a(n)| is the number of sets of ordered pairs of n labeled elements. - Steven Finch, Nov 14 2021
|a(n)| is the number of square roots of any permutation in S_{2n} whose disjoint cycle decomposition consists of n transpositions, n > 0. For n=2, permutation (1,2)(3,4) in S_4 has exactly |a(2)|=2 square roots: (1,3,2,4) and (1,4,2,3). - Luis Manuel Rivera Martínez, Feb 25 2015
Self-convolution gives A076729(n)*(-1)^n interleaved with zeros. - Vladimir Reshetnikov, Oct 11 2016
Named after the French mathematician Charles Hermite (1822-1901). - Amiram Eldar, Jun 06 2021

			From _Steven Finch_, Nov 14 2021: (Start)
|a(4)| = 12 because the sets of ordered pairs for n = 4 are
  {(1,2),(3,4)}, {(2,1),(3,4)}, {(1,2),(4,3)}, {(2,1),(4,3)},
  {(1,3),(2,4)}, {(3,1),(2,4)}, {(1,3),(4,2)}, {(3,1),(4,2)},
  {(1,4),(3,2)}, {(4,1),(3,2)}, {(1,4),(2,3)}, {(4,1),(2,3)}. (End)

G. C. Greubel, Table of n, a(n) for n = 0..730
Giuseppe Dattoli, Subuhi Khan, and Ujair Ahmad, Hermite numbers and new families of polynomials, arXiv:2503.14930 [math.NT], 2025.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Hermite Number.
Wikipedia, Hermite number.

Cf. A097388 (same sequence without zeros).
Cf. A001813, A076729, A126869, A081123.
Cf. A101109 (ordered triples instead of ordered pairs).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

A067994 := n -> pochhammer(-n, n/2):
seq(A067994(n), n = 0..31); # Peter Luschny, Nov 14 2021

HermiteH[Range[0,50], 0]
With[{nmax=50}, CoefficientList[Series[Exp[-x^2], {x,0,nmax}],x]*Range[0, nmax]!] (* G. C. Greubel, Jun 09 2018 *)

a(n) = polhermite(n, 0); \\ Michel Marcus, Feb 27 2015

x='x+O('x^30); Vec(serlaplace(exp(-x^2))) \\ G. C. Greubel, Jun 09 2018

E.g.f.: exp(-x^2). - Vladeta Jovovic, Aug 24 2002
a(n) = (-1)^(n/2)*n!/(n/2)! if n is even, 0 otherwise. - Mitch Harris, Feb 01 2006
a(n) = -(2*n-2)*a(n-2). - Alexander Karpov, Jul 24 2017
E.g.f.: U(0) where U(k) = 1 - x^2/((2*k+1) - x^2*(2*k+1)/(x^2 - 2*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: 1/G(0) where G(k) = 1 + 2*x^2*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: E(0)/(1+x) where E(k) = 1 + x/(1 - x/(x - (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: E(0)-1, where E(k) = 2 - x^2/(2*k+1 + x^2/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
a(2*k) = A097388(k), a(2*k+1) = 0. - Joerg Arndt, Oct 12 2016
From Peter Luschny, Nov 14 2021: (Start)
a(n) = A057077(n)*A126869(n)*A081123(n). In particular, a(n) is divisible by floor(n/2)!.
a(n) = Pochhammer(-n, n/2). (End)

A081124 Binomial transform of floor(n/2)!.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 90, 224, 585, 1594, 4520, 13288, 40409, 126782, 409646, 1360512, 4637681, 16202034, 57941164, 211860488, 791272129, 3015807254, 11719800674, 46401584096, 187039192185, 767058993386, 3198568491792, 13553864902504

Offset: 0

Paul Barry, Mar 07 2003

G. C. Greubel, Table of n, a(n) for n = 0..880
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Cf. A081123, A004526.

Cf. A084261.

Table[Sum[Binomial[n,k]*Floor[k/2]!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2013 *)

for(n=0,50, print1(sum(k=0,n, binomial(n,k)*(floor(k/2))!), ", ")) \\ G. C. Greubel, Feb 02 2017

a(n) = Sum_{k=0..n} C(n, k)*floor(k/2)!.
E.g.f.: exp(x)*(1+sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2)). - Vladeta Jovovic, Sep 25 2003
Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-1)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ sqrt(Pi*n)/2 * exp(sqrt(2*n)-n/2-1/2)*(n/2)^(n/2) * (1+5/(3*sqrt(2*n))). - Vaclav Kotesovec, Aug 15 2013

A211374 Product of all the parts in the partitions of n into exactly 2 parts.

Original entry on oeis.org

1, 1, 2, 12, 24, 360, 720, 20160, 40320, 1814400, 3628800, 239500800, 479001600, 43589145600, 87178291200, 10461394944000, 20922789888000, 3201186852864000, 6402373705728000, 1216451004088320000, 2432902008176640000, 562000363888803840000

Offset: 1

Wesley Ivan Hurt, Feb 06 2013

			Define a(1):=1; a(2) = 1 since 2 = 1+1 and (1)*(1) = 1; a(3) = 2 since 3 = 2+1 and (2)*(1) = 2; a(4) = 12 since 4 = 3+1 = 2+2 and (3)*(1)*(2)*(2) = 12; a(5) = 24 since 5 = 4+1 = 3+2 and (4)*(1)*(3)*(2) = 24.

G. C. Greubel, Table of n, a(n) for n = 1..449
Index entries for sequences related to partitions.

Cf. A001318, A002620, A081123, A081125, A093353.

Bisections: A002674, A010050.

[(Factorial(n-1) * Factorial(Floor(n/2)))/Factorial(n-1-Floor(n/2)) : n in [1..25]]; // Wesley Ivan Hurt, Oct 16 2014

A211374:=n->( (n-1)! * floor(n/2)! )/( (n-1) - floor(n/2) )!: seq(A211374(k), k=1..25);
with(combinat, numbperm): seq(numbperm(k-1, floor(k/2))*floor(k/2)!, k = 1..25); # Wesley Ivan Hurt, Jun 07 2013

Table[Times @@ Flatten[Select[Partitions[n], Length[#] == 2 &]], {n, 25}] (* T. D. Noe, Feb 11 2013 *)
Table[((n - 1)!*Floor[n/2]!)/(n - 1 - Floor[n/2])!, {n, 25}] (* Wesley Ivan Hurt, Oct 16 2014 *)

a(n) = prod(i=1, n\2, i*(n-i)); \\ Michel Marcus, Nov 14 2017

a(n) = ( (n-1)! * floor(n/2)! )/( n-1-floor(n/2) )!.
a(n) = P(n-1, floor(n/2)) * floor(n/2)!, where P(n,k) are the k-permutations of n objects. - Wesley Ivan Hurt, Jun 07 2013
a(2n) = A002674(n); a(2n+1) = A010050(n). - Wesley Ivan Hurt, Oct 16 2014
a(n) = Product_{i=1..floor(n/2)} i * (n-i). - Wesley Ivan Hurt, Nov 14 2017
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - cosh(1). (End)

A368338 Number T(n,k) of partitions of [n] whose sum of block maxima minus block minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 2, 1, 4, 9, 12, 12, 8, 6, 1, 5, 14, 25, 34, 36, 36, 28, 18, 6, 1, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24, 1, 7, 27, 70, 139, 224, 318, 408, 490, 534, 536, 468, 378, 258, 162, 96, 24, 1, 8, 35, 104, 237, 440, 710, 1032, 1398, 1764, 2094, 2296, 2364, 2220, 1962, 1584, 1242, 816, 528, 192, 120

Offset: 0

Alois P. Heinz, Dec 21 2023

			T(4,0) = 1: 1|2|3|4.
T(4,1) = 3: 12|3|4, 1|23|4, 1|2|34.
T(4,2) = 5: 123|4, 12|34, 13|2|4, 1|234, 1|24|3.
T(4,3) = 4: 1234, 124|3, 134|2, 14|2|3.
T(4,4) = 2: 13|24, 14|23.
T(5,5) = 8: 124|35, 125|34, 13|245, 13|25|4, 145|23, 15|23|4, 14|2|35, 15|2|34.
T(5,6) = 6: 134|25, 135|24, 14|235, 15|234, 14|25|3, 15|24|3.
T(6,9) = 6: 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,  4,  2;
  1, 4,  9, 12, 12,   8,   6;
  1, 5, 14, 25, 34,  36,  36,  28,  18,   6;
  1, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24;
  ...

Alois P. Heinz, Rows n = 0..33, flattened
Wikipedia, Partition of a set

Columns k=0..3 give: A000012, A001477(n-1), A000096(n-2), A000297(n-4).
Row sums give A000110.
Cf. A002620, A081123, A367450, A367850, A368401.

b:= proc(n, m) option remember; `if`(n=0, x^add(-i, i=m), add(
      b(n-1, subs(j=n, m)), j=m)+expand(b(n-1, {m[], n})*x^n))
    end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..(h-> h*(n-h))(iquo(n, 2))), n=0..10);
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
      b(n-1, s)*(k+1), 0)+`if`(n>k+1, b(n-1, {s[], n}), 0)+
      add(expand(x^(h-n)*b(n-1, s minus {h})), h=s))(nops(s)))
    end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);

Sum_{k=0..A002620(n)} k * T(n,k) = A367850(n).

T(n,A002620(n)) = A081123(n+1).

A231601 Number of permutations of [n] avoiding ascents from odd to even numbers.

Original entry on oeis.org

1, 1, 1, 4, 8, 54, 162, 1536, 6144, 75000, 375000, 5598720, 33592320, 592950960, 4150656720, 84557168640, 676457349120, 15620794116480, 140587147048320, 3628800000000000, 36288000000000000, 1035338990313196800, 11388728893445164800, 355902198372945100800

Offset: 0

Alois P. Heinz, Nov 11 2013

nonn

			a(0) = 1: ().
a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 4: 132, 213, 231, 321.
a(4) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
a(5) = 54: 13245, 13254, 13524, ..., 54213, 54231, 54321.
a(6) = 162: 132465, 132546, 132645, ..., 654213, 654231, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Column k=0 of A231777.
Bisection gives: A061711 (even part).
Cf. A081123, A110138, A199660, A285672.

a:= n-> ceil(n/2)!*ceil(n/2)^floor(n/2):
seq(a(n), n=0..30);

a(n) = ceiling(n/2)! * ceiling(n/2)^floor(n/2).

a(n) = A081123(n+1) * A110138(n).

A249128 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 6, 11, 7, 8, 1, 1, 6, 18, 26, 10, 11, 1, 1, 24, 50, 46, 58, 14, 15, 1, 1, 24, 96, 154, 86, 102, 18, 19, 1, 1, 120, 274, 326, 444, 156, 177, 23, 24, 1, 1, 120, 600, 1044, 756, 954, 246, 272, 28, 29, 1, 1, 720, 1764, 2556, 3708, 1692, 2016, 384, 416, 34, 35, 1, 1

Offset: 0

Clark Kimberling, Oct 22 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A056953(n) for n >= 0.
Column 1 consists of repeated factorials (A000142), as in A081123.

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (1 + x + x^2)/(1 + x), so that p(2,x) = 1 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  1    1    1
  2    3    1    1
  2    4    5    1    1
  6    11   7    8    1    1

Clark Kimberling, Rows 0..100, flattened

Cf. A056953, A000142, A081123, A249130.

z = 15; p[x_, n_] := x + Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249128 array *)
Flatten[CoefficientList[u, x]] (* A249128 sequence *)

A056043 Let k be largest number such that k^2 divides n!; a(n) = k/floor(n/2)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 30, 30, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 28, 28, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 126

Offset: 1

Labos Elemer, Jul 25 2000

nonn

			For n = 7, 7! = 5040 = 144*35, so 12 is its largest square-root-divisor, A000188(5040), and it is divisible by 6 = 3!, so a(7) = 12/3! = 2.

Cf. A000142, A000188, A001405, A004526, A055772, A081123.

f[p_, e_] := p^Floor[e/2]; b[1] = 1; a[n_] := (Times @@ f @@@ FactorInteger[n!]) / Floor[n/2]!; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

a(n) = A000188(n!)/floor(n/2)! = A055772(n)/A000142(A004526(n)) = A055772(n)/A081123(n). [Corrected by Amiram Eldar, May 24 2024]

A211229 Matrix inverse of lower triangular array A211226.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, -2, 1, -1, 1, 0, 0, -1, 1, 2, -3, 3, 0, 0, -3, 1, -2, 2, -3, 3, 0, 0, -1, 1, 9, -8, 8, -12, 6, 0, 0, -4, 1, -9, 9, -8, 8, -6, 6, 0, 0, -1, 1, 44, -45, 45, -40, 20, -30, 10, 0, 0, -5, 1, -44, 44, -45, 45, -20, 20, -10, 10, 0, 0, -1, 1

Offset: 0

Peter Bala, Apr 05 2012

This triangle is related to the derangement numbers. The subtriangles (T(2*n,2*k))n,k>=0, -(T(2*n+1,2*k))n,k>=0, and (T(2*n+1,2*k+1))n,k>=0 are all equal to A008290, while the subtriangle (T(2*n,2*k+1))n,k>=0 equals -A180188 (with an extra initial row of zeros).

			Triangle begins:
   n\k |    0    1    2    3    4    5    6    7    8    9
  =====+==================================================
    0  |    1
    1  |   -1    1
    2  |    0   -1    1
    3  |    0    0   -1    1
    4  |    1    0    0   -2    1
    5  |   -1    1    0    0   -1    1
    6  |    2   -3    3    0    0   -3    1
    7  |   -2    2   -3    3    0    0   -1    1
    8  |    9   -8    8  -12    6    0    0   -4    1
    9  |   -9    9   -8    8   -6    6    0    0   -1    1
  ...

Cf. A000166, A000240, A000387, A000449, A000475, A008290, A180188, A211226.

b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
Table[(-1)^(n+k) (b[n]!/b[k]!) Sum[(-1)^i/i!, {i, 0, b[n-k]-h}], {n, 0, 31}, {k, 0, n}] //Flatten (* Manfred Boergens, Jan 10 2023 *)
(* Sum-free code *)
b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
T[n_, k_] = (-1)^(n+k) (b[n]!/b[k]!) If[n-k<2, 1, Round[(b[n-k]-h)!/E]/(b[n-k]-h)!];
Table[T[n, k], {n, 0, 31}, {k, 0, n}] // Flatten
(* Manfred Boergens, Jan 10 2023 *)

f(n) = (n\2)!; \\ A081123
T(n,k) = f(n)/(f(k)*f(n-k)); \\ A211226
tabl(nn) = my(m=matrix(nn, nn, n, k, if (n>=k, T(n-1,k-1), 0))); 1/m; \\ Michel Marcus, Jan 10 2023

T(2*n,2*k) = T(2*n+1,2*k+1) = -T(2*n+1,2*k) = binomial(n,k)*A000166(n-k) = (n!/k!)*Sum_{i = 0..n-k} (-1)^i/i!;
T(2*n,2*k+1) = -n*binomial(n-1,k)*A000166(n-k-1) = -(n!/k!)*Sum_{i = 0..n-k-1} (-1)^i/i!.
T(n,k) = T(n-k,0)*A211226(n,k).
Column entries:
T(2*n,0) = A000166(n), T(2*n,2) = A000240(n), T(2*n,4) = A000387(n), T(2*n,6) = A000449(n), T(2*n,8) = A000475(n).
From Manfred Boergens, Jan 10 2023: (Start)
With b(j) = floor(j/2); h = 1 for n even and k odd, h = 0 else:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*Sum_{i = 0..b(n-k)-h} (-1)^i/i!.
Sum-free formula:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!) for n-k < 2.
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*round((b(n-k)-h)!/exp(1))/(b(n-k)-h)!) otherwise. (End)

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