A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.

-2, 3, 3, -7, -12, -7, 25, 48, 48, 25, -121, -260, -240, -260, -121, 721, 1830, 1500, 1500, 1830, 721, -5041, -15162, -13230, -8400, -13230, -15162, -5041, 40321, 141176, 142296, 70560, 70560, 142296, 141176, 40321, -362881, -1451592, -1695456, -874944, -423360, -874944, -1695456, -1451592, -362881

Offset

1,1

Links

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

Formula

T(n, k) = t(n, k) + t(n, n-k+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).

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

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

T(n, k) = (-1)^n * ( k! + (n-k+1)! ) * A001263(n, k).

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

Example

Triangle begins as:
     -2;
      3,      3;
     -7,    -12,     -7;
     25,     48,     48,    25;
   -121,   -260,   -240,  -260,   -121;
    721,   1830,   1500,  1500,   1830,    721;
  -5041, -15162, -13230, -8400, -13230, -15162,  -5041;
  40321, 141176, 142296, 70560,  70560, 142296, 141176, 40321;

Mathematica

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

Prog

(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))
flatten([[A169653(n, k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 23 2021
(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)) >;
[A169653(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

Crossrefs

Cf. A000262, A001263, A008297, A105278, A169154.

Sequence in context: A095978 A370361 A156763 * A129012 A136122 A121875

Adjacent sequences: A169650 A169651 A169652 * A169654 A169655 A169656

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 05 2010

Extensions

Edited by G. C. Greubel, Feb 23 2021

Status

approved