OFFSET
1,2
FORMULA
T(n,n-2) = A014106(n-1).
Conjecture: Sum_{m=0..n} T(n,m) = A126931(n). - R. J. Mathar, Mar 25 2010
T(n,k) = (2*k*sum(j=0..n+k, binomial(j,-n-3*k+2*j)*binomial(n+k,j)))/(n+k). - Vladimir Kruchinin, Oct 12 2011
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-2)^i. - Philippe Deléham, Feb 23 2012
EXAMPLE
The triangle starts
1;
2, 1;
5, 4, 1;
12, 14, 6, 1;
30, 44, 27, 8, 1;
MAPLE
MATHEMATICA
a53121[n_, k_] /; n < k || OddQ[n - k] = 0;
a53121[n_, k_] := (k + 1) Binomial[n + 1, (n - k)/2]/(n + 1);
a38207[n_, k_] := Sum[Binomial[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := Sum[a53121[n, j] a38207[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)
PROG
(Maxima)
T(n, k):=(2*k*sum(binomial(j, -n-3*k+2*j)*binomial(n+k, j), j, 0, n+k))/(n+k); /* Vladimir Kruchinin, Oct 12 2011 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jun 19 2004
EXTENSIONS
Edited, definition rewritten, program and more terms added by R. J. Mathar, Mar 25 2010
STATUS
approved