A176305 Triangle T(n,k) = 1 -A002627(k) -A002627(n-k) +A002627(n), read by rows.

1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 31, 36, 31, 1, 1, 165, 194, 194, 165, 1, 1, 1031, 1194, 1218, 1194, 1031, 1, 1, 7423, 8452, 8610, 8610, 8452, 7423, 1, 1, 60621, 68042, 69066, 69200, 69066, 68042, 60621, 1, 1, 554249, 614868, 622284, 623284, 623284, 622284, 614868, 554249, 1

Offset: 0

Author

Roger L. Bagula, Apr 14 2010

Keywords

• nonn
• tabl

Comments

Row sums are {1, 2, 4, 16, 100, 720, 5670, 48972, 464660, 4829372, 54711782, ...}.

Row sums s(n) appear to obey (2-n)*s(n) +(n+2)*(n-1)*s(n-2) -n*(2*n-1)*s(n-2) +n*(n-2)*s(n-3)=0. - R. J. Mathar, May 04 2013

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      2,      1;
  1,      7,      7,      1;
  1,     31,     36,     31,      1;
  1,    165,    194,    194,    165,      1;
  1,   1031,   1194,   1218,   1194,   1031,      1;
  1,   7423,   8452,   8610,   8610,   8452,   7423,      1;
  1,  60621,  68042,  69066,  69200,  69066,  68042,  60621,      1;
  1, 554249, 614868, 622284, 623284, 623284, 622284, 614868, 554249, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

Magma

b:= func< n | Factorial(n)*(Exp(1)-1)>;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return 1 +Floor(b(n)) -Floor(b(k)) -Floor(b(n-k));
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019

Maple

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else 1 +floor(n!*(exp(1)-1)) -floor(k!*(exp(1)-1)) -floor((n-k)!*(exp(1)-1))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019

Mathematica

(* First program *)
a[n_]:= a[n] = If[n==0,0,n*a[n-1] +1];
T[n_, k_]:= T[n, k] = 1 -(a[k] +a[n-k] -a[n]);
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, 1 +Floor[n!*(E-1)] -Floor[k!*(E-1)] - Floor[(n-k)!*(E-1)]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 26 2019 *)

PARI

T(n,k) = if(k==0 || k==n, 1, 1 +floor(n!*(exp(1)-1)) -floor(k!*(exp(1)-1)) -floor((n-k)!*(exp(1)-1)) ); \\ G. C. Greubel, Nov 26 2019

Sage

@CachedFunction
def b(n): return factorial(n)*(exp(1)-1);
def T(n, k):
    if (k==0 or k==n): return 1
    else: return 1 +floor(b(n)) -floor(b(k)) -floor(b(n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019

Formula

T(n,k) = T(n,n-k).