OFFSET
0,1
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
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.
EXAMPLE
-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;
...
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Jan 18 2009
EXTENSIONS
Partially edited by Andrew Howroyd, Mar 26 2020
Additionally edited by G. C. Greubel, May 28 2020
STATUS
approved