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

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, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1, 1, 4, -22, -24, 85, 14, -112, 42

Offset: 0

Author

Ralf Stephan, Sep 12 2004

Keywords

• sign
• tabl

Comments

Also, coefficients of polynomials that have values in A098495 and A094954.

Examples

			Triangle begins:
   1;
   0, -1;
  -1, -1, 1;
  -1,  1, 2, -1;
   0,  3, 0, -3, 1;
  ...
		

Links

• Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Crossrefs

Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.

Cf. A098494 (diagonal polynomials), A085478, A244419.

Programs

Maple

A098493 := proc (n, k)
add((-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), j = k..n);
end proc:
seq(seq(A098493(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

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)))))

Formula

T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].

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).

G.f.: (1-x)/(1+(y-1)*x+x^2). [Vladeta Jovovic, Dec 14 2009]

From Peter Bala, Jul 13 2021: (Start)

Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).

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)

From Peter Bala, Jun 26 2025: (Start)

n-th row polynomial R(n, x) = Sum_{k = 0..n} (-1)^k * binomial(n+k, 2*k) * (1 + x)^k.

R(n, 2*x + 1) = (-1)^n * Dir(n, x), where Dir(n,x) denotes the n-th row polynomial of the triangle A244419.

R(n, -1 - x) = b(n, x), where b(n, x) denotes the n-th row polynomial of the triangle A085478. (End)