A154987 Triangle read by rows: T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n!/k!)*(2*(n + k) - 1).

-2, 4, 4, 13, 20, 13, 41, 69, 69, 41, 183, 268, 264, 268, 183, 1099, 1405, 1080, 1080, 1405, 1099, 7943, 9486, 5970, 4080, 5970, 9486, 7943, 65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547, 604831, 685672, 384552, 149520, 77280, 149520, 384552, 685672, 604831

Offset: 0

Author

Roger L. Bagula, Jan 18 2009

Keywords

• sign
• tabl

Examples

			     -2;
      4,     4;
     13,    20,    13;
     41,    69,    69,    41;
    183,   268,   264,   268,   183;
   1099,  1405,  1080,  1080,  1405,  1099;
   7943,  9486,  5970,  4080,  5970,  9486,  7943;
  65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Programs

Maple

t:= proc(n,k) option remember; ## simplified t;
2*(n+k-1/2)*(n!/k!);
end proc:
A154987:= proc(n,k) ## n >= 0 and k = 0 .. n
t(n,k) + t(n,n-k)
end proc: # Yu-Sheng Chang, Apr 13 2020

Mathematica

(* First program *)
t[n_, k_]:= 2*n!*Gamma[n+k+1/2]/(k!*Gamma[n+k-1/2]);
T[n_, k_]:= t[n, k] + t[n,n-k];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
(* Second Program *)
T[n_, k_]:= Binomial[n, k]*((n-k)!*(2*n+2*k-1) + k!*(4*n-2*k-1));
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 28 2020 *)

Sage

def T(n, k): return binomial(n, k)*(factorial(n-k)*(2*n+2*k-1) + factorial(k)*(4*n-2*k-1))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 28 2020

Formula

T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*n!*Gamma(n + k + 1/2)/(k!*Gamma(n + k - 1/2)).

T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n+k-1/2)*(n!/k!). - Yu-Sheng Chang, Apr 13 2020

From G. C. Greubel, May 28 2020: (Start)

T(n,k) = binomial(n,k)*( (2*n+2*k-1)*(n-k)! + (4*n-2*k-1)*k! ).

T(n,n-k) = T(n,k), for k >= 0.

Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*n!*e_{n-1}(1) ), where e_{n}(x) is the finite exponential function = Sum_{k=0..n} x^k/k!.

Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*A007526(n) ).

T(n,0) = A175925(n-1) + 2*n.

T(n,1) = A007680(n) + A001107(n). (End)

Extensions

Partially edited by Andrew Howroyd, Mar 26 2020

Additionally edited by G. C. Greubel, May 28 2020