A126325 Triangle read by rows: T(n,k) = binomial(2*n+1, n-k) (1 <= k <= n).

1, 5, 1, 21, 7, 1, 84, 36, 9, 1, 330, 165, 55, 11, 1, 1287, 715, 286, 78, 13, 1, 5005, 3003, 1365, 455, 105, 15, 1, 19448, 12376, 6188, 2380, 680, 136, 17, 1, 75582, 50388, 27132, 11628, 3876, 969, 171, 19, 1, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1

Offset

1,2

Comments

T(n,k) is the total area between the lines y=k-1 and y=k in all Dyck paths of semilength n (1 <= k <= n).

With row and column indices starting at 0, this triangle is the Riordan array ( c(x)^4/(2 - c(x)), x*c^2(x) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Mar 12 2022

Equals A111418 when k starts at 0. - Georg Fischer, Jul 26 2023

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for n >= 2, k >= 2.

T(n,1) = A002054(n); T(n,2) = A003516(n); T(n,3) = A030053(n);

T(n,4) = A030054(n); T(n,5) = A030055(n).

Row sums yield A008549.

Example

Triangle begins:
     1;
     5,    1;
    21,    7,    1;
    84,   36,    9,    1;
   330,  165,   55,   11,    1;
  1287,  715,  286,   78,   13,    1;
  5005, 3003, 1365,  455,  105,   15,    1;
  ..

Maple

T:=(n, k)->binomial(2*n+1, n-k): for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Mathematica

t[n_, k_] := Binomial[2n + 1, n - k]; Table[ t[n, k], {n, 10}, {k, n}] // Flatten

Prog

(PARI) for(n=1, 15, for(k=1, n, print1(binomial(2*n+1, n-k), ", "))) \\ G. C. Greubel, Oct 23 2018
(Magma) [[Binomial(2*n+1, n-k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 23 2018
(GAP) T:=Flat(List([1..10], n->List([1..n], k->Binomial(2*n+1, n-k)))); # Muniru A Asiru, Oct 24 2018

Crossrefs

Cf. A000108, A008549, A002054, A003516, A030053, A030054, A030055, A111418.

Sequence in context: A296306 A101693 A063476 * A278544 A095368 A029757

Adjacent sequences: A126322 A126323 A126324 * A126326 A126327 A126328

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 11 2007

Status

approved