This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A101950 #51 May 02 2025 08:01:04
%S A101950 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 A101950 -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 A101950 -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 A101950 Product of A049310 and A007318 as lower triangular matrices.
%C A101950 A Chebyshev and Pascal product.
%C A101950 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 A101950 Apart from signs, identical with A104562.
%C A101950 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 A101950 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
%C A101950 Also the convolution triangle of the inverse of 6th cyclotomic polynomial A010892. - _Peter Luschny_, Oct 08 2022
%H A101950 Vincenzo Librandi, <a href="/A101950/b101950.txt">Table of n, a(n) for n = 0..1325</a>
%H A101950 Jerry Ray Dias, <a href="https://hrcak.srce.hr/102724">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].
%H A101950 Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, <a href="https://doi.org/10.1016/j.jmaa.2016.04.033">Root geometry of polynomial sequences. II: Type (1,0)</a>, J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).
%F A101950 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 A101950 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 A101950 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 A101950 G.f.: 1/(1-x-x^2-y*x). - _Philippe Deléham_, Feb 10 2012
%F A101950 T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - _Philippe Deléham_, Mar 07 2014
%F A101950 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 A101950 Triangle begins:
%e A101950 1,
%e A101950 1, 1,
%e A101950 0, 2, 1,
%e A101950 -1, 1, 3, 1,
%e A101950 -1,-2, 3, 4, 1,
%e A101950 ...
%e A101950 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 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
%p A101950 # Uses function PMatrix from A357368. Adds a row on top and a column to the left.
%p A101950 PMatrix(10, n -> [0, 1, 1, 0, -1,-1][irem(n, 6) + 1]); # _Peter Luschny_, Oct 08 2022
%t A101950 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 A101950 Cf. A049310, A007318, A104562, A010892, A084938.
%K A101950 easy,sign,tabl
%O A101950 0,5
%A A101950 _Paul Barry_, Dec 22 2004
%E A101950 Typo in formula corrected and information added by _Johannes W. Meijer_, Aug 06 2011