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

1, 1, 1, 3, 6, 3, 10, 30, 30, 10, 35, 140, 210, 140, 35, 126, 630, 1260, 1260, 630, 126, 462, 2772, 6930, 9240, 6930, 2772, 462, 1716, 12012, 36036, 60060, 60060, 36036, 12012, 1716, 6435, 51480, 180180, 360360, 450450, 360360, 180180, 51480

Offset

0,4

Comments

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

Links

Indranil Ghosh, Rows 0..100, flattened

Formula

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).

It appears that T(n, k) = A088218(n)*binomial(n, k). - Michel Marcus, Jun 11 2015

Example

[1]
[1,1]
[3,6,3]
[10,30,30,10]
[35,140,210,140,35]

Mathematica

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 *)
Flatten[Table[Binomial[n+k-1, k] Binomial[2n-1, n-k], {n, 0, 9}, {k, 0, n}]] (* Indranil Ghosh, Mar 04 2017 *)

Prog

(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
(PARI)
tabl(nn) = {for (n=0, nn, for(k=0, n, print1(binomial(n+k-1, k)*binomial(2*n-1, n-k), ", "); ); print(); ); };
tabl(9); \\ Indranil Ghosh, Mar 04 2017

Crossrefs

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

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

Sequence in context: A210193 A368952 A120906 * A210201 A275535 A160899

Adjacent sequences: A258755 A258756 A258757 * A258759 A258760 A258761

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Jun 10 2015

Status

approved