A105937 Infinite square array read by antidiagonals: T(m, 0) = 1, T(m, 1) = m; T(m, k) = (m - k + 1) T(m+1, k-1) - (k-1) (m+1) T(m+2, k-2).

1, 1, 0, 1, 1, -2, 1, 2, -2, 0, 1, 3, 0, -12, 36, 1, 4, 4, -24, 24, 0, 1, 5, 10, -30, -60, 420, -1800, 1, 6, 18, -24, -216, 720, -720, 0, 1, 7, 28, 0, -420, 420, 5040, -30240, 176400, 1, 8, 40, 48, -624, -960, 14400, -40320, 40320, 0, 1, 9, 54, 126, -756, -3780, 22680, 22680, -589680, 3764880, -28576800

Offset

0,6

References

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

Links

G. C. Greubel, Antidiagonals n = 0..100, flattened

Formula

See A127080 for e.g.f.

Example

Array begins
   1  1  1   1   1   1   1   1   1   1 ... (A000012)
   0  1  2   3   4   5   6   7   8   9 ... (A001477)
  -2 -2  0   4  10  18  28  40  54  70 ... (A028552)
   0 12 24  30  24   0  48 126 240 396 ... (A126935)
  36 24 60 216 420 624 756 720 396 360 ... (A126958)
...

Maple

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=1 then n
    else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 28 2020

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 28 2020 *)

Prog

(PARI) T(n, k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) )); \\ G. C. Greubel, Jan 28 2020
(Magma)
function T(n, k)
  if k eq 0 then return 1;
  elif k eq 1 then return n;
  else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
  end if; return T; end function;
[T(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 28 2020
(Sage)
@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==1): return n
    else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 28 2020

Crossrefs

Rows give A000027, A028552, A126935, A126958.

Columns give A126934, A126962, A127067, A127068, A127070.

A127080 gives another version of the array.

Sequence in context: A122864 A140084 A243747 * A035146 A035216 A258587

Adjacent sequences: A105934 A105935 A105936 * A105938 A105939 A105940

Keyword

sign,tabl

Author

Vincent v.d. Noort, Mar 24 2007

Extensions

More terms added by G. C. Greubel, Jan 28 2020

Status

approved