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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).
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
0,1
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)
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
Sequence in context: A086915 A059927 A290437 * A089419 A341235 A363649
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