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

0,1

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.

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

Adjacent sequences: A154984 A154985 A154986 * A154988 A154989 A154990

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