OFFSET
0,8
COMMENTS
Reflected version of A106566.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.
FORMULA
T(n, k) = A106566(n, n-k).
Sum_{k=0..n} T(n,k) = A000108(n).
T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - Jean-François Alcover, Jun 14 2019
Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - G. C. Greubel, Jun 14 2022
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.
EXAMPLE
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 2, 2, 0;
1, 3, 5, 5, 0;
1, 4, 9, 14, 14, 0;
1, 5, 14, 28, 42, 42, 0;
1, 6, 20, 48, 90, 132, 132, 0;
1, 7, 27, 75, 165, 297, 429, 429, 0;
1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 0;
1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 0;
...
MATHEMATICA
T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 14 2019 *)
PROG
(Sage)
@CachedFunction
def A130020(n, k):
if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
return add(A130020(n-1, j) for j in (0..k))
for n in (0..10) :
[A130020(n, k) for k in (0..n)] # Peter Luschny, Nov 14 2012
(Magma)
A130020:= func< n, k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
[A130020(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2022
(PARI) {T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* Michael Somos, Oct 01 2022 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Jun 16 2007
STATUS
approved