OFFSET
0,4
COMMENTS
Row sums are A109262(n+1).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1} = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022
EXAMPLE
Triangle begins:
1;
1, 1;
2, 3, 1;
5, 9, 4, 1;
14, 28, 14, 6, 1;
42, 90, 48, 27, 7, 1;
132, 297, 165, 110, 35, 9, 1;
Production matrix is:
1...1
1...2...1
0...1...1...1
0...1...1...2...1
0...0...0...1...1...1
0...0...0...1...1...2...1
0...0...0...0...0...1...1...1
0...0...0...0...0...1...1...2...1
0...0...0...0...0...0...0...1...1...1
0...0...0...0...0...0...0...1...1...2...1
0...0...0...0...0...0...0...0...0...1...1...1
...
MATHEMATICA
T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)
PROG
(Magma)
F:=Factorial;
A236843:= func< n, k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022
(SageMath)
F=factorial
def A236843(n, k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022
(PARI) T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Feb 01 2014
STATUS
approved