|
| |
|
|
A101950
|
|
Product of A049310 and A007318 as lower triangular matrices.
|
|
46
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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]
|
|
|
LINKS
| 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].
|
|
|
FORMULA
| Number triangle 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). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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). - DELEHAM Philippe, Feb 10 2012
|
|
|
EXAMPLE
| Rows begin {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 ; ... [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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]
|
|
|
CROSSREFS
| Cf. A104562.
Sequence in context: A196660 A135222 A124094 * A104562 A164306 A111603
Adjacent sequences: A101947 A101948 A101949 * A101951 A101952 A101953
|
|
|
KEYWORD
| easy,sign,tabl,changed
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Dec 22 2004
|
|
|
EXTENSIONS
| Typo in formula corrected and information added by Johannes W. Meijer, Aug 06 2011
|
| |
|
|
