A201701 - OEIS

%I #34 Dec 25 2022 13:58:53

%S 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,

%T 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,

%U 512,1280,1120,400,50,1,0,0,0,0,0

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

%C 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.

%C Skewed version of triangle in A200139.

%C Triangle without zeros: A207537.

%C 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

%C This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - _Peter Bala_, Jul 14 2015

%H C. Corsani, D. Merlini, and R. Sprugnoli, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00110-6">Left-inversion of combinatorial sums</a>, Discrete Mathematics, 180 (1998) 107-122.

%F 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 .

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

%F 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.

%F G.f.: (1-x)/(1-2*x-y*x^2). - _Philippe Deléham_, Mar 03 2012

%F From _Peter Bala_, Jul 14 2015: (Start)

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

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

%e The triangle T(n,k) begins :

%e n\k      0     1     2     3     4    5   6  7 8 9 10 11 12 13 14 15 ...

%e 0:       1

%e 1:       1     0

%e 2:       2     1     0

%e 3:       4     3     0     0

%e 4:       8     8     1     0     0

%e 5:      16    20     5     0     0    0

%e 6:      32    48    18     1     0    0   0

%e 7:      64   112    56     7     0    0   0  0

%e 8:     128   256   160    32     1    0   0  0 0

%e 9:     256   576   432   120     9    0   0  0 0 0

%e 10:    512  1280  1120   400    50    1   0  0 0 0  0

%e 11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0

%e 12:   2048  6144  6912  3584   840   72   1  0 0 0  0  0  0

%e 13:   4096 13312 16640  9984  2912  364  13  0 0 0  0  0  0  0

%e 14:   8192 28672 39424 26880  9408 1568  98  1 0 0  0  0  0  0  0

%e 15:  16384 61440 92160 70400 28800 6048 560 15 0 0  0  0  0  0  0  0

%e ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

%e -------------------------------------------------------------------------

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)

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

%Y Diagonals sums are in A052980.

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

%Y Cf. A098158, A200139, A207537.

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

%K nonn,easy,tabl

%O 0,4

%A _Philippe Deléham_, Dec 03 2011

%E Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by _Wolfdieter Lang_, Aug 06 2014