OFFSET
0,2
COMMENTS
The main diagonal and column 0 of the triangle are the central binomial coefficients, which are the sums of the squares of Pascal's triangle entries. This sum representation can be generalized, and all terms can be seen as sums of coefficients of some polynomials. (See the Example section.)
FORMULA
EXAMPLE
Triangle starts:
[0] 1;
[1] 2, 2;
[2] 6, 9, 6;
[3] 20, 40, 40, 20;
[4] 70, 175, 225, 175, 70;
[5] 252, 756, 1176, 1176, 756, 252;
[6] 924, 3234, 5880, 7056, 5880, 3234, 924;
[7] 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432;
.
Because of the symmetry, only the sum representation of terms with k <= n/2 are shown.
0: [1]
1: [1+1]
2: [1+4+1], [1+4+4]
3: [1+9+9+1], [1+9+21+9]
4: [1+16+36+16+1], [1+16+66+76+16], [1+16+76+96+36]
5: [1+25+100+100+25+1], [1+25+160+340+205+25], [1+25+190+460+400+100]
MAPLE
T := (n, k) -> binomial(k + n, k) * binomial(2*n - k, n):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
MATHEMATICA
T[n_, k_] := Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, -n +k, 1, 1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Mar 21 2024
STATUS
approved