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A098493
Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.
8
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
0,9
COMMENTS
Also, coefficients of polynomials that have values in A098495 and A094954.
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.
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)
EXAMPLE
Triangle begins:
1;
0, -1;
-1, -1, 1;
-1, 1, 2, -1;
0, 3, 0, -3, 1;
...
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
PROG
(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)))))
CROSSREFS
Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.
Cf. A098494 (diagonal polynomials), A085478, A244419.
Sequence in context: A058558 A210869 A123973 * A058560 A131047 A366548
KEYWORD
sign,tabl
AUTHOR
Ralf Stephan, Sep 12 2004
STATUS
approved