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
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
Programs
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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
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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
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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
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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
Comments