OFFSET
0,4
COMMENTS
row sums = A135231
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = (3+(-1)^n)/2 and T(n,n) = 1. - G. C. Greubel, Nov 20 2019
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
1, 2, 2, 1;
2, 2, 4, 3, 1;
1, 3, 6, 7, 4, 1;
2, 3, 9, 13, 11, 5, 1;
1, 4, 12, 22, 24, 16, 6, 1;
2, 4, 16, 34, 46, 40, 22, 7, 1;
...
MAPLE
T:= proc(n, k) option remember;
if k=n then 1
elif k=0 then (3+(-1)^n)/2
else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}] ]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)
PROG
(PARI) T(n, k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019
(Magma)
function T(n, k)
if k eq n then return 1;
elif k eq 0 then return (3+(-1)^n)/2;
else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
end if; return T; end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
(Sage)
@CachedFunction
def T(n, k):
if (k==n): return 1
elif (k==0): return (3+(-1)^n)/2
else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 23 2007
EXTENSIONS
More terms and offset changed by G. C. Greubel, Nov 20 2019
STATUS
approved