A123162 - OEIS

%I #30 Jul 19 2023 07:57:07

%S 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,

%T 1,1,13,286,1287,1716,715,78,1,1,15,455,3003,6435,5005,1365,105,1,1,

%U 17,680,6188,19448,24310,12376,2380,136,1,1,19,969,11628,50388,92378,75582,27132,3876,171,1

%N Triangle read by rows: T(n,k) = binomial(2*n - 1, 2*k - 1) for 0 < k <= n and T(n,0) = 1.

%H Muniru A Asiru, <a href="/A123162/b123162.txt">Rows n = 0..50, flattened</a>

%F From _Paul Barry_, May 26 2008: (Start)

%F T(n,k) = binomial(2*n - 1, 2*k - 1) + 0^k.

%F 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)

%F From _Franck Maminirina Ramaharo_, Oct 10 2018: (Start)

%F 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).

%F 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).

%F 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)

%F From _G. C. Greubel_, Jul 18 2023: (Start)

%F Sum_{k=0..n} T(n,k) = A123166(n).

%F T(n, n-1) = (n-1)*T(n, 1), n >= 2.

%F T(2*n, n) = A259557(n).

%F T(2*n+1, n+1) = A002458(n). (End)

%e Triangle begins:

%e      1;

%e      1,  1;

%e      1,  3,   1;

%e      1,  5,  10,    1;

%e      1,  7,  35,   21,    1;

%e      1,  9,  84,  126,   36,    1;

%e      1, 11, 165,  462,  330,   55,    1;

%e      1, 13, 286, 1287, 1716,  715,   78,  1;

%e      1, 15, 455, 3003, 6435, 5005, 1365, 105, 1;

%e      ...

%t T[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

%o (Maxima) T(n, k) := if k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$

%o tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* _Franck Maminirina Ramaharo_, Oct 10 2018 */

%o (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

%o (SageMath)

%o def A123162(n,k): return binomial(2*n-1, 2*k-1) + int(k==0)

%o flatten([[A123162(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 01 2022

%o (Magma)

%o A123162:= func< n,k | k eq 0 select 1 else Binomial(2*n-1, 2*k-1) >;

%o [A123162(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 18 2023

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

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Oct 02 2006

%E Edited by _N. J. A. Sloane_, Oct 04 2006

%E Partially edited and offset corrected by _Franck Maminirina Ramaharo_, Oct 10 2018