A176022 Triangle T(n, k) = A176013(n, k) + A176013(n, n-k+1), read by rows.

-2, 3, 3, -7, -18, -7, 25, 96, 96, 25, -121, -650, -800, -650, -121, 721, 5490, 7500, 7500, 5490, 721, -5041, -53067, -92610, -73500, -92610, -53067, -5041, 40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321, -362881, -6532164, -20345472, -18373824, -10668672, -18373824, -20345472, -6532164, -362881

Offset

1,1

Comments

Row sums are: -2, 6, -32, 242, -2342, 27422, -374936, 5841922, -101897354, 1962916022, ...

Links

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

Formula

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

From G. C. Greubel, Feb 15 2021: (Start)

T(n, k) = A176013(n, k) + A176013(n, n-k+1), where A176013(n, k) = (-1)^n*(n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1).

Sum_{k=1..n} T(n, k) = 2*(-1)^n * n! * Hypergeometric2F2(-n, -(n-1); 2, 2; 1). (End)

Example

Triangle begins as:
     -2;
      3,      3;
     -7,    -18,      -7;
     25,     96,      96,     25;
   -121,   -650,    -800,   -650,   -121;
    721,   5490,    7500,   7500,   5490,     721;
  -5041, -53067,  -92610, -73500, -92610,  -53067,  -5041;
  40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321;

Mathematica

(* First program *)
T[n_, m_]:= ((-1)^n*n!/(m*m!))*Binomial[n-1, m-1]*Binomial[n, m-1] + ((-1)^n*n!)/((n-m+1)*(n-m+1)!)*Binomial[n-1, n-m] Binomial[n, n-m];
Table[T[n, m], {m, n}], {n, 10}]//Flatten
(* Second program *)
A176013[n_, k_] := (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, k_]:= A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

Prog

(Sage)
def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 15 2021
(Magma)
A176013:= func< n, k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 15 2021

Crossrefs

Cf. A008297, A176013, A176021.

Sequence in context: A136122 A121875 A036251 * A316275 A113031 A127582

Adjacent sequences: A176019 A176020 A176021 * A176023 A176024 A176025

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 06 2010

Extensions

Edited by G. C. Greubel, Feb 15 2021

Status

approved