A117317 Triangle related to partitions of n.

1, 2, 1, 4, 5, 1, 8, 16, 9, 1, 16, 44, 41, 14, 1, 32, 112, 146, 85, 20, 1, 64, 272, 456, 377, 155, 27, 1, 128, 640, 1312, 1408, 833, 259, 35, 1, 256, 1472, 3568, 4712, 3649, 1652, 406, 44, 1, 512, 3328, 9312, 14608, 14002, 8361, 3024, 606, 54, 1, 1024, 7424, 23552

Offset: 0

Author

Paul Barry, Mar 07 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A007052. Diagonal sums are A052988. Reversal of A056242.

Essentially given by (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 28 2012

Examples

			Triangle begins
1,
2, 1,
4, 5, 1,
8, 16, 9, 1,
16, 44, 41, 14, 1,
32, 112, 146, 85, 20, 1,
64, 272, 456, 377, 155, 27, 1
Triangle (0, 2, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, 1/2, 0, 0, ...) begins :
1
0, 1
0, 2, 1
0, 4, 5, 1
0, 8, 16, 9, 1
0, 16, 44, 41, 14, 1
0, 32, 112, 146, 85, 20, 1
0, 64, 272, 456, 377, 155, 27, 1
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Crossrefs

Cf. Columns : A000079, A053220, A056243 ; Diagonals : A000012, A000096

Programs

Haskell

a117317 n k = a117317_tabl !! n !! k
a117317_row n = a117317_tabl !! n
a117317_tabl = map reverse a056242_tabl
-- Reinhard Zumkeller, May 08 2014

Formula

Number triangle T(n,k)=sum{j=0..n-k, C(n+j,k)C(n-k,j)}

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) for n>1. - Philippe Deléham, Jan 28 2012

G.f.: (1-y*x)/(1-2*(y+1)*x+y*(y+1)*x^2). - Philippe Deléham, Jan 28 2012