A265376 -id:A265376 - cp's OEIS frontend

A356013 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)).

1, 1, 1, 2, 3, 1, 6, 6, 4, 1, 24, 30, 20, 5, 1, 120, 120, 60, 30, 6, 1, 720, 840, 420, 210, 42, 7, 1, 5040, 5040, 3360, 840, 336, 56, 8, 1, 40320, 45360, 20160, 7560, 3024, 504, 72, 9, 1, 362880, 362880, 201600, 75600, 15120, 5040, 720, 90, 10, 1

Offset: 1

Seiichi Manyama, Jul 23 2022

			Triangle begins:
      1;
      1,     1;
      2,     3,     1;
      6,     6,     4,    1;
     24,    30,    20,    5,    1;
    120,   120,    60,   30,    6,   1;
    720,   840,   420,  210,   42,   7,  1;
   5040,  5040,  3360,  840,  336,  56,  8, 1;
  40320, 45360, 20160, 7560, 3024, 504, 72, 9, 1;
  ...

Row sums gives A356011.
Column k=1..3 give A000142(n-1), |A265376(n)|, A356012.
Cf. A355996.

PARI
```
T(n, k) = n!/(k!*(n\k));
```

E.g.f. of column k: -(1 - x^k) * log(1 - x^k)/(k! * (1 - x)).

A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

Original entry on oeis.org

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000

Offset: 0

Peter McNair, Mar 19 2022

nonn

Nimsum 2*k + 2 = A004443(2*k).
Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.
Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.
Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).
a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Cf. A004443, A092605, A265376, A334367.

a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)

PARI
```
a(n) = 2*prod(k=2,n,bitxor(2*k, 2))
```

a(n) = 2*Product_{k=2..n} A004443(2*k).
a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.
a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.
a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.

cp's OEIS Frontend

A356013 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)).

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