|
%I #19 May 28 2020 03:04:40
%S -2,4,4,13,20,13,41,69,69,41,183,268,264,268,183,1099,1405,1080,1080,
%T 1405,1099,7943,9486,5970,4080,5970,9486,7943,65547,75775,43806,20370,
%U 20370,43806,75775,65547,604831,685672,384552,149520,77280,149520,384552,685672,604831
%N 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).
%H G. C. Greubel, <a href="/A154987/b154987.txt">Rows n = 0..100 of the triangle, flattened</a>
%F 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)).
%F 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
%F From _G. C. Greubel_, May 28 2020: (Start)
%F T(n,k) = binomial(n,k)*( (2*n+2*k-1)*(n-k)! + (4*n-2*k-1)*k! ).
%F T(n,n-k) = T(n,k), for k >= 0.
%F 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!.
%F Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*A007526(n) ).
%F T(n,0) = A175925(n-1) + 2*n.
%F T(n,1) = A007680(n) + A001107(n). (End)
%e -2;
%e 4, 4;
%e 13, 20, 13;
%e 41, 69, 69, 41;
%e 183, 268, 264, 268, 183;
%e 1099, 1405, 1080, 1080, 1405, 1099;
%e 7943, 9486, 5970, 4080, 5970, 9486, 7943;
%e 65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547;
%e ...
%p t:= proc(n,k) option remember; ## simplified t;
%p 2*(n+k-1/2)*(n!/k!);
%p end proc:
%p A154987:= proc(n,k) ## n >= 0 and k = 0 .. n
%p t(n,k) + t(n,n-k)
%p end proc: # _Yu-Sheng Chang_, Apr 13 2020
%t (* First program *)
%t t[n_, k_]:= 2*n!*Gamma[n+k+1/2]/(k!*Gamma[n+k-1/2]);
%t T[n_, k_]:= t[n, k] + t[n,n-k];
%t Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
%t (* Second Program *)
%t T[n_, k_]:= Binomial[n, k]*((n-k)!*(2*n+2*k-1) + k!*(4*n-2*k-1));
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 28 2020 *)
%o (Sage)
%o def T(n, k): return binomial(n, k)*(factorial(n-k)*(2*n+2*k-1) + factorial(k)*(4*n-2*k-1))
%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 28 2020
%K sign,tabl
%O 0,1
%A _Roger L. Bagula_, Jan 18 2009
%E Partially edited by _Andrew Howroyd_, Mar 26 2020
%E Additionally edited by _G. C. Greubel_, May 28 2020
|
