OFFSET
1,2
COMMENTS
Triangle T(n,k)=
1. Riordan Array (1,(1-sqrt(1-4*x))/((1-x)*2)) without first column.
2. Riordan Array ((1-sqrt(1-4*x))/((1-x)*2*x),(1-sqrt(1-4*x))/((1-x)*2)) numbering triangle (0,0).
Convolution triangle of A014137(n). - Philippe Deléham, Jan 23 2014
FORMULA
T(n,k):=k*sum(i=0..n-k, (binomial(i+k-1,k-1)*binomial(2*(n-i)-k-1,n-i-1))/(n-i)).
EXAMPLE
Triangle:
1,
2, 1,
4, 4, 1,
9, 12, 6, 1,
23, 34, 24, 8, 1,
65, 98, 83, 40, 10, 1,
197, 294, 273, 164, 60, 12, 1
MATHEMATICA
T[n_, k_]:= (k/n) (Binomial[-1 - k + 2 n, -1 + n] HypergeometricPFQ[{k, k - n, -n}, {1/2 + k/2 - n, 1 + k/2 - n}, 1/4]);
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // TableForm (* Peter Luschny, May 30 2022 *)
PROG
(Maxima)
T(n, k):=k*sum((binomial(i+k-1, k-1)*binomial(2*(n-i)-k-1, n-i-1))/(n-i), i, 0, n-k);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Nov 25 2011
STATUS
approved