A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

1, 2, 0, 3, 4, 0, 4, 16, 0, 0, 5, 40, 16, 0, 0, 6, 80, 96, 0, 0, 0, 7, 140, 336, 64, 0, 0, 0, 8, 224, 896, 512, 0, 0, 0, 0, 9, 336, 2016, 2304, 256, 0, 0, 0, 0, 10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0, 11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Offset

0,2

Comments

Row n is the sum of the convolution of A089627(p,i) with A089627(n-p,i), for p=0,1,...,n.

Links

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

Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.

Formula

T(n,k) = 4^k * C(n+1, 2*k+1).

T(n,k) = sum(p=0..n, sum(i=0..k, C(p,i)*C(p-i, i)*C(n-p,k-i)*C(n-p-k+i, k-i))).

T(n,k) = A085841(n/2,k), if n is even.

T(n,k) = 2^k * A105070(n,k).

Example

Triangle:
1
2, 0
3, 4, 0
4, 16, 0, 0
5, 40, 16, 0, 0
6, 80, 96, 0, 0, 0
7, 140, 336, 64, 0, 0, 0
8, 224, 896, 512, 0, 0, 0, 0
9, 336, 2016, 2304, 256, 0, 0, 0, 0
10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0
11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Crossrefs

Sequence in context: A368090 A209705 A181289 * A352835 A349953 A349339

Adjacent sequences: A229029 A229030 A229031 * A229033 A229034 A229035

Keyword

nonn,tabl

Author

Rui Duarte and António Guedes de Oliveira, Sep 11 2013

Status

approved