A098495 Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r >= 0.

1, 1, 0, 1, -1, -1, 1, -2, -1, -1, 1, -3, 1, 1, 0, 1, -4, 5, 1, 1, 1, 1, -5, 11, -7, -2, -1, 1, 1, -6, 19, -29, 9, 1, -1, 0, 1, -7, 29, -71, 76, -11, 1, 1, -1, 1, -8, 41, -139, 265, -199, 13, -2, 1, -1, 1, -9, 55, -239, 666, -989, 521, -15, 1, -1, 0, 1, -10, 71, -377, 1393, -3191, 3691, -1364, 17, 1, -1, 1, 1, -11, 89, -559

Offset

0,8

Links

Table of n, a(n) for n=0..81.

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

Recurrence: T(r, 1) = 1, T(r, 2) = -r-1, T(r, c) = -rT(r, c-1) - T(r, c-2). (Corrected Oct 19 2004)

Example

Array begins
  1,  0, -1,  -1,   0,    1,    1,   0, -1, ...
  1, -1, -1,   1,   1,   -1,   -1,   1,  1, ...
  1, -2,  1,   1,  -2,    1,    1,  -2,  1, ...
  1, -3,  5,  -7,   9,  -11,   13, -15, ...
  1, -4, 11, -29,  76, -199,  521, ...
  1, -5, 19, -71, 265, -989, 3691, ...
  ...

Mathematica

T[r_, 1] := 1; T[r_, 2] := -r - 1; T[r_, c_] := -r*T[r, c - 1] - T[r, c - 2]; Flatten[ Table[ T[n - i, i], {n, 0, 11}, {i, n + 1}]] (* Robert G. Wilson v, May 10 2005 *)

Prog

(PARI) { t(r, c)=if(c>r||c<0||r<0, 0, if(c>=r-1, (-1)^r*if(c==r, 1, -c), if(r==1, 0, if(c==0, t(r-1, 0)-t(r-2, 0), t(r-1, c)-t(r-2, c)-t(r-1, c-1))))) } T(r, c)=sum(i=0, c, t(c, i)*r^i)

Crossrefs

See A094954 (with negative k) for negative r and more formulas and programs.

Rows include (-1)^c times A005408, A002878, A001834, A030221, A002315. Columns include A028387. Antidiagonal sums are in A098496.

Sequence in context: A125692 A128258 A104967 * A175432 A204118 A095025

Adjacent sequences: A098492 A098493 A098494 * A098496 A098497 A098498

Keyword

sign,tabl

Author

Ralf Stephan, Sep 12 2004

Extensions

More terms from Robert G. Wilson v, May 10 2005

Status

approved