A156006 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.

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, 138, 152, 138, 138, 152, 138, 1, 1, 436, 456, 372, 330, 372, 456, 436, 1, 1, 1438, 1465, 1111, 847, 847, 1111, 1465, 1438, 1, 1, 4871, 4906, 3586, 2431, 2002, 2431, 3586, 4906, 4871, 1

Offset: 0

Author

Roger L. Bagula, Feb 01 2009

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins as:
  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,  138,  152,  138,  138,  152,  138,    1;
  1,  436,  456,  372,  330,  372,  456,  436,    1;
  1, 1438, 1465, 1111,  847,  847, 1111, 1465, 1438,    1;
  1, 4871, 4906, 3586, 2431, 2002, 2431, 3586, 4906, 4871, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A009799, A068875, A128634.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return ((n-k)/n)*Binomial(n+k-1, k) + (k/(n-k))*Binomial(2*n-k-1, n);
    fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) )));

Magma

function T(n,k)
  if k eq n then return 1;
  else return ((n-k)/n)*Binomial(n+k-1, k) + (k/(n-k))*Binomial(2*n-k-1, n);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 02 2019

Maple

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

Mathematica

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

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

Sage

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return ((n-k)/n)*binomial(n+k-1, k) + (k/(n-k))*binomial(2*n-k-1, n)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 02 2019

Formula

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.

From G. C. Greubel, Dec 02 2019: (Start)

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.

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

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

Extensions

Edited by G. C. Greubel, Dec 02 2019