A097750 Reversal of the binomial transform of the Whitney triangle A004070 (see A131250), triangle read by rows, T(n,k) for 0 <= k <= n.

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 8, 22, 26, 16, 1, 10, 37, 64, 57, 32, 1, 12, 56, 130, 163, 120, 64, 1, 14, 79, 232, 386, 382, 247, 128, 1, 16, 106, 378, 794, 1024, 848, 502, 256, 1, 18, 137, 576, 1471, 2380, 2510, 1816, 1013, 512, 1, 20, 172, 834, 2517, 4944, 6476, 5812, 3797, 2036, 1024

Offset

0,3

Comments

Reversal of the Riordan array (1/(1-2x), x/(1-x)^2), see A131250. Row sums are A061667 and diagonal sums of A131250 are A045623. The n-th row elements correspond to the end elements of the 2n-th row of the Whitney triangle A004070. A131250 corresponds to the product of Pascal's triangle and the Whitney triangle.

Links

Table of n, a(n) for n=0..65.

Formula

T(n, k) = Sum_{i=0..n} binomial(n+k, i-k).

T(n, k) = T(n-1,k)+2*T(n-1,k-1)-T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014

T(n, k) = binomial(2*n-k, k)*hypergeom([1, 1, -k], [1, 1 - 2*k + 2*n], -1). - Peter Luschny, Oct 28 2018

Example

Triangle begins:
1;
1, 2;
1, 4, 4;
1, 6, 11, 8;
1, 8, 22, 26, 16;
1, 10, 37, 64, 57, 32;
1, 12, 56, 130, 163, 120, 64;
1, 14, 79, 232, 386, 382, 247, 128;

Maple

T := (n, k) -> binomial(2*n-k, k)*hypergeom([1, 1, -k], [1, 1-2*k+2*n], -1):
for n from 0 to 8 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Oct 28 2018

Mathematica

T[_, 0] = 1; T[n_, n_] := 2^n; T[n_, k_] /; 0 < k < n := T[n, k] = T[n - 1, k] + 2 T[n - 1, k - 1] - T[n - 2, k - 2]; T[_, _] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 19 2019 *)

Crossrefs

Row sums are A061667.

Cf. A004070, A097761, A131250.

Sequence in context: A200057 A136600 A136672 * A304623 A133544 A303872

Adjacent sequences: A097747 A097748 A097749 * A097751 A097752 A097753

Keyword

easy,nonn,tabl

Author

Paul Barry, Aug 23 2004

Extensions

Definition and comments corrected by Philippe Deléham, Jan 11 2014

Status

approved