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A101950 Product of A049310 and A007318 as lower triangular matrices. 56

%I

%S 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,

%T -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,

%U -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

%N Product of A049310 and A007318 as lower triangular matrices.

%C A Chebyshev and Pascal product.

%C 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)).

%C Apart from signs, identical with A104562.

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A101950/b101950.txt">Table of n, a(n) for n = 0..1325</a>

%H J. R. Dias, <a href="http://www.stkpula.hr/ccacaa/CCA-PDF/cca2004/v77-n1_n2/CCA_77_2004_325-330_dias.pdf"> Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons </a>, Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].

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

%F 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

%F 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].

%F G.f.: 1/(1-x-x^2-y*x). - _Philippe Deléham_, Feb 10 2012

%F T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - _Philippe Deléham_, Mar 07 2014

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

%e Triangle begins:

%e 1,

%e 1,1,

%e 0,2,1,

%e -1,1,3,1,

%e -1,-2,3,4,1,

%e ..

%e 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

%p 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

%t 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_ *)

%Y Cf. A104562.

%K easy,sign,tabl

%O 0,5

%A _Paul Barry_, Dec 22 2004

%E Typo in formula corrected and information added by _Johannes W. Meijer_, Aug 06 2011

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Last modified November 14 19:12 EST 2018. Contains 317214 sequences. (Running on oeis4.)