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A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

%I A098493 #30 Jun 30 2025 23:46:11
%S A098493 1,0,-1,-1,-1,1,-1,1,2,-1,0,3,0,-3,1,1,2,-5,-2,4,-1,1,-2,-7,6,5,-5,1,
%T A098493 0,-5,0,15,-5,-9,6,-1,-1,-3,12,9,-25,1,14,-7,1,-1,3,15,-18,-29,35,7,
%U A098493 -20,8,-1,0,7,0,-42,14,63,-42,-20,27,-9,1,1,4,-22,-24,85,14,-112,42
%N A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.
%C A098493 Also, coefficients of polynomials that have values in A098495 and A094954.
%H A098493 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.
%F A098493 T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].
%F A098493 Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
%F A098493 G.f.: (1-x)/(1+(y-1)*x+x^2). [_Vladeta Jovovic_, Dec 14 2009]
%F A098493 From _Peter Bala_, Jul 13 2021: (Start)
%F A098493 Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).
%F A098493 T(n,k) = (-1)^k * the (n,k)-th entry of Q^(-1)*P = Sum_{j = k..n} (-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), where P denotes Pascal's triangle A007318 and Q denotes triangle A061554 (formed from P by sorting the rows into descending order). (End)
%F A098493 From _Peter Bala_, Jun 26 2025: (Start)
%F A098493 n-th row polynomial R(n, x) = Sum_{k = 0..n} (-1)^k * binomial(n+k, 2*k) * (1 + x)^k.
%F A098493 R(n, 2*x + 1) = (-1)^n * Dir(n, x), where Dir(n,x) denotes the n-th row polynomial of the triangle A244419.
%F A098493 R(n, -1 - x) = b(n, x), where b(n, x) denotes the n-th row polynomial of the triangle A085478. (End)
%e A098493 Triangle begins:
%e A098493    1;
%e A098493    0, -1;
%e A098493   -1, -1, 1;
%e A098493   -1,  1, 2, -1;
%e A098493    0,  3, 0, -3, 1;
%e A098493   ...
%p A098493 A098493 := proc (n, k)
%p A098493 add((-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), j = k..n);
%p A098493 end proc:
%p A098493 seq(seq(A098493(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Jul 13 2021
%o A098493 (PARI) T(n,k)=if(k>n||k<0||n<0,0,if(k>=n-1,(-1)^n*if(k==n,1,-k),if(n==1,0,if(k==0,T(n-1,0)-T(n-2,0),T(n-1,k)-T(n-2,k)-T(n-1,k-1)))))
%Y A098493 Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.
%Y A098493 Cf. A098494 (diagonal polynomials), A085478, A244419.
%K A098493 sign,tabl
%O A098493 0,9
%A A098493 _Ralf Stephan_, Sep 12 2004