A156006 - OEIS

%I #10 Sep 08 2022 08:45:41

%S 1,1,1,1,2,1,1,4,4,1,1,8,10,8,1,1,18,23,23,18,1,1,47,56,56,56,47,1,1,

%T 138,152,138,138,152,138,1,1,436,456,372,330,372,456,436,1,1,1438,

%U 1465,1111,847,847,1111,1465,1438,1,1,4871,4906,3586,2431,2002,2431,3586,4906,4871,1

%N Triangle, read by rows, T(n, k) = ((n-k)/(n+k))*binomial(n+k, n) + (k/(2*n-k))*binomial(2*n -k, n), with T(0,0) = 1.

%C Row sums are A068875(n): {1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, ...}.

%H G. C. Greubel, <a href="/A156006/b156006.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = ((n-k)/(n+k))*binomial(n+k, n) + (k/(2*n-k))*binomial(2*n -k, n), with T(0,0) = 1.

%F From _G. C. Greubel_, Dec 02 2019: (Start)

%F T(n, k) = ((n-k)/n)*binomial(n+k-1, k) + (k/(n-k))*binomial(2*n-k-1, n), with T(n,n) = 1.

%F Sum_{k=0..n} T(n, k) = A068875(n).

%F Sum_{k=1..n-1} T(n,k) = A128634(n), n >= 1. (End)

%e Triangle begins as:

%e   1;

%e   1,    1;

%e   1,    2,    1;

%e   1,    4,    4,    1;

%e   1,    8,   10,    8,    1;

%e   1,   18,   23,   23,   18,    1;

%e   1,   47,   56,   56,   56,   47,    1;

%e   1,  138,  152,  138,  138,  152,  138,    1;

%e   1,  436,  456,  372,  330,  372,  456,  436,    1;

%e   1, 1438, 1465, 1111,  847,  847, 1111, 1465, 1438,    1;

%e   1, 4871, 4906, 3586, 2431, 2002, 2431, 3586, 4906, 4871, 1;

%p seq(seq( `if`(k=n, 1, ((n-k)/n)*binomial(n+k-1, k) + (k/(n-k))*binomial(2*n-k-1, n)), k=0..n), n=0..10); # _G. C. Greubel_, Dec 02 2019

%t T[n_, k_]:= If[n==0, 1, ((n-k)/(n+k))*Binomial[n+k, n] + (k/(2*n-k))*Binomial[2*n -k, n]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (PARI) T(n,k) = if(k==n, 1, ((n-k)/n)*binomial(n+k-1, k) + (k/(n-k))*binomial(2*n-k-1, n) ); \\ _G. C. Greubel_, Dec 02 2019

%o (Magma)

%o function T(n,k)

%o   if k eq n then return 1;

%o   else return ((n-k)/n)*Binomial(n+k-1, k) + (k/(n-k))*Binomial(2*n-k-1, n);

%o   end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 02 2019

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k==n): return 1

%o     else: return ((n-k)/n)*binomial(n+k-1, k) + (k/(n-k))*binomial(2*n-k-1, n)

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 02 2019

%o (GAP)

%o T:= function(n,k)

%o     if k=n then return 1;

%o     else return ((n-k)/n)*Binomial(n+k-1, k) + (k/(n-k))*Binomial(2*n-k-1, n);

%o     fi; end;

%o Flat(List([1..15], n-> List([1..n], k-> T(n,k) )));

%Y Cf. A009799, A068875, A128634.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 01 2009

%E Edited by _G. C. Greubel_, Dec 02 2019