A123162 Triangle read by rows: T(n,k) = binomial(2*n - 1, 2*k - 1) for 0 < k <= n and T(n,0) = 1.
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, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1
Offset: 0
Examples
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;
...
Links
- Muniru A Asiru, Rows n = 0..50, flattened
Programs
-
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
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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
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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 -
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 */
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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
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)
Extensions
Edited by N. J. A. Sloane, Oct 04 2006
Partially edited and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018