%I #38 Sep 08 2022 08:46:12

%S 1,1,1,3,6,3,10,30,30,10,35,140,210,140,35,126,630,1260,1260,630,126,

%T 462,2772,6930,9240,6930,2772,462,1716,12012,36036,60060,60060,36036,

%U 12012,1716,6435,51480,180180,360360,450450,360360,180180,51480

%N Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).

%C Triangle T(n,k), read by rows, given by (1, 2, 1/2, 3/2, 2/3, 4/3, 3/4, 5/4, ...) DELTA (1, 2, 1/2, 3/2, 2/3, 4/3, 3/4, 5/4, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jun 19 2015

%H Indranil Ghosh, <a href="/A258758/b258758.txt">Rows 0..100, flattened</a>

%F G.f.: A(x) = 1/(2 - C(x+y)), where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. of Catalan numbers (A000108).

%F It appears that T(n, k) = A088218(n)*binomial(n, k). - _Michel Marcus_, Jun 11 2015

%e [1]

%e [1,1]

%e [3,6,3]

%e [10,30,30,10]

%e [35,140,210,140,35]

%t max = 10; s = (2*(x + y))/(-1 + 4*x + Sqrt[1 - 4*x - 4*y] + 4*y) + O[x]^(max+2) + O[y]^(max+2); t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[t[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2015, after _Vladimir Kruchinin_ *)

%t Flatten[Table[Binomial[n+k-1,k] Binomial[2n-1,n-k], {n, 0, 9}, {k, 0, n}]] (* _Indranil Ghosh_, Mar 04 2017 *)

%o (Magma) [[Binomial(n+k-1,k)*Binomial(2*n-1,n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jun 12 2015

%o (PARI)

%o tabl(nn) = {for (n=0, nn, for(k=0, n, print1(binomial(n+k-1,k)*binomial(2*n-1,n-k),", "););print(););};

%o tabl(9); \\ _Indranil Ghosh_, Mar 04 2017

%Y Columns k=0-1 give: A088218, A002457(n-1) for n>0.

%Y Cf. A069723 (row sums, with a shift).

%K nonn,tabl

%O 0,4

%A _Vladimir Kruchinin_, Jun 10 2015