A127068 Let d(m, 0) = 1, d(m, 1) = m, and d(m, k) = (m - k + 1)*d(m+1, k-1) - (k-1)*(m+1) d(m+2, k-2). Sequence gives d(3,n).

1, 3, 4, -30, -216, 420, 14400, 22680, -1411200, -8482320, 195955200, 2399997600, -36883123200, -788107320000, 9066542284800, 318173519664000, -2824576634880000, -159078423407904000, 1088403529973760000, 97970873094110016000, -508476519708917760000, -73631427647097640320000

Offset: 0

Author

Vincent v.d. Noort, Mar 21 2007

Keywords

• sign

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150

Crossrefs

A column of A105937.

Programs

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(3, n): n in [0..25]]; // G. C. Greubel, Jan 29 2020

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(T(3, n), n=0..25); # G. C. Greubel, Jan 29 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[3, n], {n,0,25}] (* G. C. Greubel, Jan 29 2020 *)

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) ));
vector(25, n, T(3, (n-1)) ) \\ G. C. Greubel, Jan 29 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(3, n) for n in (0..25)] # G. C. Greubel, Jan 29 2020

Formula

From Peter Bala, Feb 15 2022: (Start)

Conjectures:

a(2*n) = (-1)^(n+1)*(n + 1)*(2*n - 1)*(2*n)!.

a(2*n+1) = - 2*(2*n + 3)*(3*n - 2)*a(2*n-1) - 4*(n - 1)*(2*n + 3)*(4*n^2 - 1)*a(2*n-3) with a(1) = 3 and a(3) = -30. (End)