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

0,5

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

Cf. A002458, A007318, A034867, A103327, A122366, A123166, A259557.

Sequence in context: A086385 A295222 A362078 * A213998 A340970 A294946

Adjacent sequences: A123159 A123160 A123161 * A123163 A123164 A123165

Keyword

nonn,tabl,easy

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