A026022 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(n,k) = C(n,k) for k = 1,2,...,n, for n = 1,2,3; and for n >= 4, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[ (n+2)/2 ] and T(n,(n+3)/2) = T(n-1,(n+1)/2) if n is odd.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 5, 10, 10, 4, 1, 6, 15, 20, 14, 1, 7, 21, 35, 34, 14, 1, 8, 28, 56, 69, 48, 1, 9, 36, 84, 125, 117, 48, 1, 10, 45, 120, 209, 242, 165, 1, 11, 55, 165, 329, 451, 407, 165, 1, 12, 66, 220, 494, 780, 858, 572, 1, 13, 78, 286, 714, 1274, 1638, 1430, 572

Offset

1,5

References

E. Lucas, Théorie des Nombres, Albert Blanchard, Paris, 1958,tome1, p.88

Links

Table of n, a(n) for n=1..75.

Formula

T(n, k) = C(n, k) - C(n, k-4). - Ralf Stephan, Jan 09 2005

T(2n,n) = A026029(n). - Philippe Deléham, Mar 12 2013

T(2n-1,n) = A026016(n), n>0. - Philippe Deléham, Mar 12 2013

Example

From Philippe Deléham, Mar 12 2013: (Start)
Triangle begins:
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 4, 6, 4
1, 5, 10, 10, 4
1, 6, 15, 20, 14
1, 7, 21, 35, 34, 14
1, 8, 28, 56, 69, 48
1, 9, 36, 84, 125, 117, 48
1, 10, 45, 120, 209, 242, 165
1, 11, 55, 165, 329, 451, 407, 165
Pentagon arithmetic of Delannoy (in E. Lucas):
1, 1, 1, 1, 0
1, 2, 3, 4, 4, 0
1, 3, 6, 10, 14, 14, 0
1, 4, 10, 20, 34, 48, 48, 0
1, 5, 15, 35, 69, 117, 165, 165,
1, 6, 21, 56, 125, 242, 407, 572,
1, 7, 28, 84, 209, 451, 858, 1430  (End)

Prog

(PARI) {T(n, k) = if( 2*k < n+4, binomial( n, k) - binomial( n, k-4), 0)} /* Michael Somos, Jan 08 2012 */

Crossrefs

Sequence in context: A296115 A118687 A281587 * A073714 A171848 A144151

Adjacent sequences: A026019 A026020 A026021 * A026023 A026024 A026025

Keyword

nonn,tabf

Author

Clark Kimberling

Status

approved