OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Binomial transform of the beheaded Pascal's triangle (A074909) as a matrix. (The beheaded Pascal matrix deletes the rightmost border of 1's.)
From G. C. Greubel, Aug 05 2021: (Start)
T(n, k) = Sum_{j=0..n} binomial(n, j)*binomial(j+1, k) - binomial(n, k-1), with T(n, 0) = 2^n.
T(n, k) = 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1).
Sum_{k=0..n} T(n, k) = A027649(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106515(n). (End)
EXAMPLE
First few rows of the triangle are:
1;
2, 2;
4, 7, 3;
8, 19, 15, 4
16, 47, 52, 26, 5;
32, 111, 155, 110, 40, 6;
64, 255, 426, 385, 200, 57, 7;
128, 575, 1113, 1211, 805, 329, 77, 8;
256, 1279, 2808, 3556, 2856, 1498, 504, 100, 9;
...
MAPLE
A212362 := proc(n, k)
add( binomial(n, i)*A074909(i, k), i=0..n) ;
end proc: # R. J. Mathar, Aug 03 2015
MATHEMATICA
T[n_, k_]= 2^(n-k)*Binomial[n+1, k] + (2^(n-k) -1)*Binomial[n, k-1];
Table[T[n, k] , {n, 0, 12}, {k, 0, n}] //Flatten (* G. C. Greubel, Aug 05 2021 *)
PROG
(Magma)
A074909:= func< n, k | k lt 0 or k gt n select 0 else Binomial(n+1, k) >;
[A212362(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 05 2021
(Sage)
def T(n, k): return 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 05 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jun 29 2012
EXTENSIONS
a(22) corrected by G. C. Greubel, Aug 05 2021
STATUS
approved