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A258758 Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).

%I A258758 #38 Sep 08 2022 08:46:12
%S A258758 1,1,1,3,6,3,10,30,30,10,35,140,210,140,35,126,630,1260,1260,630,126,
%T A258758 462,2772,6930,9240,6930,2772,462,1716,12012,36036,60060,60060,36036,
%U A258758 12012,1716,6435,51480,180180,360360,450450,360360,180180,51480
%N A258758 Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).
%C A258758 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 A258758 Indranil Ghosh, <a href="/A258758/b258758.txt">Rows 0..100, flattened</a>
%F A258758 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 A258758 It appears that T(n, k) = A088218(n)*binomial(n, k). - _Michel Marcus_, Jun 11 2015
%e A258758 [1]
%e A258758 [1,1]
%e A258758 [3,6,3]
%e A258758 [10,30,30,10]
%e A258758 [35,140,210,140,35]
%t A258758 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 A258758 Flatten[Table[Binomial[n+k-1,k] Binomial[2n-1,n-k], {n, 0, 9}, {k, 0, n}]] (* _Indranil Ghosh_, Mar 04 2017 *)
%o A258758 (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 A258758 (PARI)
%o A258758 tabl(nn) = {for (n=0, nn, for(k=0, n, print1(binomial(n+k-1,k)*binomial(2*n-1,n-k),", "););print(););};
%o A258758 tabl(9); \\ _Indranil Ghosh_, Mar 04 2017
%Y A258758 Columns k=0-1 give: A088218, A002457(n-1) for n>0.
%Y A258758 Cf. A069723 (row sums, with a shift).
%K A258758 nonn,tabl
%O A258758 0,4
%A A258758 _Vladimir Kruchinin_, Jun 10 2015