OFFSET
0,2
FORMULA
G.f.: (1 - t)^(-x)*(1 + t)^(2-x) = Sum_{n >= 0} R(n, x)*t^n/floor((n+1)/2)! = 1 + 2*t/1! + (1 + x)*t^2/1! + 4*x*t^3/2! + x*(3 + x)*t^4/2! + 6*x*(1 + x)*t^5/3! + x*(1 + x)*(5 + x)*t^6/3! + 8*x*(1 + x)*(2 + x)*t^7/3! + x*(1 + x)*(2 + x)*(7 + x)*t^8/4! + 10*x*(1 + x)*(2 + x)*(3 + x)*t^9/5! + ....
Row polynomials: R(2*n, x) = (2*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
R(2*n+1, x) = (2*n + 2) * Product_{i = 0..n-1} (x + i) for n >= 0.
T(2*n+1, k) = (2*n + 2)*|Stirling1(n, k)| = (2*n + 2)*A132393(n, k).
n-th row sum equals 2 * floor((n+1)/2)! for n >= 1.
EXAMPLE
Triangle begins
n\k | 0 1 2 3 4 5
- - - - - - - - - - - - - - - - -
0 | 1
1 | 2
2 | 1 1
3 | 0 4
4 | 0 3 1
5 | 0 6 6
6 | 0 5 6 1
7 | 0 16 24 8
8 | 0 14 23 10 1
9 | 0 60 110 60 10
10 | 0 54 105 65 15 1
...
MAPLE
with(combinat):
T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (n/2)*abs(Stirling1((n-2)/2, k)) else (n+1)*abs(Stirling1((n-1)/2, k)) end if; end proc:
seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Peter Bala, Apr 09 2024
STATUS
approved