A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.

1, 1, 2, 1, 8, 3, 1, 18, 25, 4, 1, 32, 98, 56, 5, 1, 50, 270, 336, 105, 6, 1, 72, 605, 1320, 891, 176, 7, 1, 98, 1183, 4004, 4719, 2002, 273, 8, 1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9, 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10, 1, 200, 5415

Offset

0,3

Comments

Sum_{k=0..n}T(n,k)*x^k = A164111(n), A000012(n), A002001(n), A001653(n+1), A001019(n), A166965(n) for x =-1, 0, 1, 2, 4, 9 respectively.

Links

Table of n, a(n) for n=0..57.

Formula

G.f.: (1-x)/(1-2*x*(1+y)+x^2*(1-y)^2).

T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)+2*T(n-2,k-1)-T(n-2,k)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.

T(n,k) = (A086645(n,k) + A091042(n,k))/2.

T(n,k) = binomial(2*n,2*k)*(2*n+1-k)/(2*n+1-2*k). - Peter Luschny, Nov 26 2013

Example

Triangle begins:
1
1, 2
1, 8, 3
1, 18, 25, 4
1, 32, 98, 56, 5
1, 50, 270, 336, 105, 6
1, 72, 605, 1320, 891, 176, 7
1, 98, 1183, 4004, 4719, 2002, 273, 8
1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9

Maple

T := (n, k) -> binomial(2*n, 2*k)*(2*n+1-k)/(2*n+1-2*k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Nov 26 2013

Mathematica

Flatten[Table[(Binomial[2n, 2k]+Binomial[2n+1, 2k])/2, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jul 05 2015 *)

Crossrefs

Cf. A086645, A091042

Cf. Columns : A000012, A001105, A180324 ; Diagonals: A000027, A131423

Cf. T(2*n,n): A228329, Row sums : A002001

Sequence in context: A011208 A001281 A331312 * A065826 A286899 A123235

Adjacent sequences: A232532 A232533 A232534 * A232536 A232537 A232538

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 25 2013

Status

approved