A101950 Product of A049310 and A007318 as lower triangular matrices.

1, 1, 1, 0, 2, 1, -1, 1, 3, 1, -1, -2, 3, 4, 1, 0, -4, -2, 6, 5, 1, 1, -2, -9, 0, 10, 6, 1, 1, 3, -9, -15, 5, 15, 7, 1, 0, 6, 3, -24, -20, 14, 21, 8, 1, -1, 3, 18, -6, -49, -21, 28, 28, 9, 1, -1, -4, 18, 36, -35, -84, -14, 48, 36, 10, 1, 0, -8, -4, 60, 50, -98, -126, 6, 75, 45, 11, 1, 1, -4, -30, 20, 145, 36, -210

Offset

0,5

Comments

A Chebyshev and Pascal product.

Row sums are n+1, diagonal sums the constant sequence 1 resp. A023434(n+1). Riordan array (1/(1-x+x^2),x/(1-x+x^2)).

Apart from signs, identical with A104562.

Subtriangle of the triangle given by [0,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010

The Fi1 and Fi2 sums lead to A004525 and the Gi1 sums lead to A077889, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 06 2011

Also the convolution triangle of the inverse of 6th cyclotomic polynomial A010892. - Peter Luschny, Oct 08 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1325

J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons , Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].

Gross, Jonathan L. ; Mansour, Toufik; Tucker, Thomas W.; Wang, David G. L. Root geometry of polynomial sequences. II: Type (1,0), J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).

Formula

T(n, k) = Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*(1+(-1)^(n+j))*C(j,k)/2.

T(0,0) = 1, T(n,k) = 0,if k>n or if k<0, T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010

p(n,x) = (x+1)*p(n-1,x)-p(n-2,x) with p(0,x) = 1 and p(1,x) = x+1 [Dias].

G.f.: 1/(1-x-x^2-y*x). - Philippe Deléham, Feb 10 2012

T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - Philippe Deléham, Mar 07 2014

T(n,k) = C(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4)) for n>=1. - Peter Luschny, Apr 25 2016

Example

Triangle begins:
1,
1,1,
0,2,1,
-1,1,3,1,
-1,-2,3,4,1,
..
Triangle [0,1,-1,1,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins : 1 ; 0,1 ; 0,1,1 ; 0,0,2,1 ; 0,-1,1,3,1 ; 0,-1,-2,3,4,1 ; ... - Philippe Deléham, Jan 27 2010

Maple

A101950 := proc(n, k) local j, k1: add((-1)^((n-j)/2)*binomial((n+j)/2, j)*(1+(-1)^(n+j))* binomial(j, k)/2, j=0..n) end: seq(seq(A101950(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 06 2011
# Uses function PMatrix from A357368. Adds a row on top and a column to the left.
PMatrix(10, n -> [0, 1, 1, 0, -1, -1][irem(n, 6) + 1]); # Peter Luschny, Oct 08 2022

Mathematica

T[0, 0] = 1; T[n_, k_] /; k>n || k<0 = 0; T[n_, k_] := T[n, k] = T[n-1, k-1]+T[n-1, k]-T[n-2, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

Crossrefs

Cf. A049310, A007318, A104562, A010892.

Sequence in context: A357137 A333381 A124094 * A104562 A164306 A309931

Adjacent sequences: A101947 A101948 A101949 * A101951 A101952 A101953

Keyword

easy,sign,tabl

Author

Paul Barry, Dec 22 2004

Extensions

Typo in formula corrected and information added by Johannes W. Meijer, Aug 06 2011

Status

approved