A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset

0,4

Comments

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Skewed version of triangle in A200139.

Triangle without zeros: A207537.

For the version with negative odd numbered columns, which is Riordan (((1-x)/(1-2*x), -x^2/(1-2*x) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014

This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

Links

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

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Formula

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1 , T(1,1) = 0 and T(n,k)=0 for k<0 or for n<k .

Sum_{k, 0<=k<=n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Example

The triangle T(n,k) begins :
n\k      0     1     2     3     4    5   6  7 8 9 10 11 12 13 14 15 ...
0:       1
1:       1     0
2:       2     1     0
3:       4     3     0     0
4:       8     8     1     0     0
5:      16    20     5     0     0    0
6:      32    48    18     1     0    0   0
7:      64   112    56     7     0    0   0  0
8:     128   256   160    32     1    0   0  0 0
9:     256   576   432   120     9    0   0  0 0 0
10:    512  1280  1120   400    50    1   0  0 0 0  0
11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
12:   2048  6144  6912  3584   840   72   1  0 0 0  0  0  0
13:   4096 13312 16640  9984  2912  364  13  0 0 0  0  0  0  0
14:   8192 28672 39424 26880  9408 1568  98  1 0 0  0  0  0  0  0
15:  16384 61440 92160 70400 28800 6048 560 15 0 0  0  0  0  0  0  0
...  reformatted and extended. - Wolfdieter Lang, Aug 06 2014
-------------------------------------------------------------------------

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Crossrefs

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

Diagonals sums are in A052980.

Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).

Cf. A098158, A200139, A207537.

Cf. A028297 (signed version, zeros deleted). Cf. A034839.

Sequence in context: A106375 A194734 A255528 * A131667 A322080 A361958

Adjacent sequences: A201698 A201699 A201700 * A201702 A201703 A201704

Keyword

nonn,easy,tabl

Author

Philippe Deléham, Dec 03 2011

Extensions

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

Status

approved