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A131238
Triangle read by rows: T(n,k) = 2*binomial(n,k) - binomial(floor((n+k)/2), k) (0 <= k <= n).
2
1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 9, 7, 1, 1, 7, 17, 16, 9, 1, 1, 9, 24, 36, 25, 11, 1, 1, 10, 36, 60, 65, 36, 13, 1, 1, 12, 46, 102, 125, 106, 49, 15, 1, 1, 13, 62, 148, 237, 231, 161, 64, 17, 1, 1, 15, 75, 220, 385, 483, 392, 232, 81, 19, 1, 1, 16, 95, 295, 625, 868, 896, 624, 321, 100, 21, 1
OFFSET
0,5
COMMENTS
Row sums = A027934: (1, 2, 5, 11, 24, 51, 107, ...).
FORMULA
T(n,k) = 2*A007318(n,k) - A046854(n,k) as infinite lower triangular matrices, where A007318 = Pascal's triangle and A046854 = Pascal's triangle with repeats, by columns.
EXAMPLE
First few rows of the triangle:
1;
1, 1;
1, 3, 1;
1, 4, 5, 1;
1, 6, 9, 7, 1;
1, 7, 17, 16, 9, 1;
1, 9, 24, 36, 25, 11, 1;
1, 10, 36, 60, 65, 36, 13, 1;
...
MAPLE
T := proc (n, k) options operator, arrow; 2*binomial(n, k)-binomial(floor((1/2)*n+(1/2)*k), k) end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jul 09 2007
MATHEMATICA
With[{B = Binomial}, Table[2*B[n, k] - B[Floor[(n+k)/2], k], {n, 0, 12}, {k, 0, n}]]//Flatten (* G. C. Greubel, Jul 12 2019 *)
PROG
(PARI) b=binomial; T(n, k) = 2*b(n, k) - b((n+k)\2, k);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 12 2019
(Magma) B:=Binomial; [2*B(n, k) - B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019
(Sage) b=binomial; [[2*b(n, k) - b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019
(GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 2*B(n, k) - B(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 12 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jun 21 2007
EXTENSIONS
More terms added by G. C. Greubel, Jul 12 2019
STATUS
approved