OFFSET
0,5
LINKS
Muniru A Asiru, Rows n = 0..50, flattened
FORMULA
From Paul Barry, May 26 2008: (Start)
T(n,k) = binomial(2*n - 1, 2*k - 1) + 0^k.
Column k has g.f. (x^k/(1 - x)^(2*k + 0^k))*Sum_{j=0..k} binomial(2*k, 2*j)*x^j. (End)
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of ((x + sqrt(x))*(sqrt(x) - 1)^(2*n) + (x - sqrt(x))*(sqrt(x) + 1)^(2*n) + 2*x - 2)/(2*x - 2).
G.f.: (1 - (2 + x)*y + (1 - 2*x)*y^2 - (x - x^2)*y^3)/(1 - (3 + 2*x)*y + (3 + x^2)*y^2 - (1 - 2*x + x^2)*y^3).
E.g.f.: ((x + sqrt(x))*exp(y*(sqrt(x) - 1)^2) + (x - sqrt(x))*exp(y*(sqrt(x) + 1)^2) + (2*x - 2)*exp(y) - 2*x)/(2*x - 2). (End)
From G. C. Greubel, Jul 18 2023: (Start)
Sum_{k=0..n} T(n,k) = A123166(n).
T(n, n-1) = (n-1)*T(n, 1), n >= 2.
T(2*n, n) = A259557(n).
T(2*n+1, n+1) = A002458(n). (End)
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 5, 10, 1;
1, 7, 35, 21, 1;
1, 9, 84, 126, 36, 1;
1, 11, 165, 462, 330, 55, 1;
1, 13, 286, 1287, 1716, 715, 78, 1;
1, 15, 455, 3003, 6435, 5005, 1365, 105, 1;
...
MATHEMATICA
T[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Maxima) T(n, k) := if k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */
(GAP) Flat(Concatenation([1], List([1..10], n->Concatenation([1], List([1..n], m->Binomial(2*n-1, 2*m-1)))))); # Muniru A Asiru, Oct 11 2018
(SageMath)
def A123162(n, k): return binomial(2*n-1, 2*k-1) + int(k==0)
flatten([[A123162(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2022
(Magma)
A123162:= func< n, k | k eq 0 select 1 else Binomial(2*n-1, 2*k-1) >;
[A123162(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Oct 02 2006
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2006
Partially edited and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018
STATUS
approved