A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.

1, 1, 1, 1, -4, 1, 1, 24, 24, 1, 1, -138, -118, -138, 1, 1, 1110, 780, 780, 1110, 1, 1, -10120, -8188, -3358, -8188, -10120, 1, 1, 100856, 101976, 30240, 30240, 101976, 100856, 1, 1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1

Offset: 1

Author

Roger L. Bagula, Apr 05 2010

Keywords

• sign
• tabl
• easy
• less

Examples

			Triangle begins as:
  1;
  1,        1;
  1,       -4,        1;
  1,       24,       24,       1;
  1,     -138,     -118,    -138,      1;
  1,     1110,      780,     780,   1110,       1;
  1,   -10120,    -8188,   -3358,  -8188,  -10120,        1;
  1,   100856,   101976,   30240,  30240,  101976,   100856,        1;
  1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710,        1;
  1, 12700890, 18147240, 9132480, 816480,  816480,  9132480, 18147240, 12700890, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000262, A001263, A008297, A105278, A169653.

Programs

Magma

A001263:= func< n,k | Binomial(n-1, k-1)*Binomial(n,k-1)/k >;
A169653:= func< n,k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
A169654:= func< n,k | A169653(n, k) - A169653(n, 1) + 1 >;
[A169654(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

Mathematica

t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
T[n_, m_] = t[n, m] + t[n, n-m+1] - (-1)^n*(n! + 1) + 1;
Table[T[n,k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Feb 23 2021 *)

Sage

def A001263(n, k): return binomial(n-1, k-1)*binomial(n,k-1)/k
def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1
flatten([[A169654(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 23 2021

Formula

T(n, k) = t(n, k) + t(n, n-k+1) - t(n, 1) - t(n, n) + 1, where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).

T(n, k) = A008297(n,k) + A008297(n,n-k+1) - (A008297(n,1) + A008297(n,n)) + 1.

From G. C. Greubel, Feb 23 2021: (Start)

T(n, k) = A169653(n, k) - A169653(n, 1) + 1

T(n, k) = A169653(n, k) - (-1)^n * (n! + 1) + 1.

T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1) - (n! + 1) + (-1)^n).

Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)

Extensions

Edited by G. C. Greubel, Feb 23 2021