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

0,5

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

Sequence in context: A166345 A015110 A128596 * A139349 A168347 A120475

Adjacent sequences: A176302 A176303 A176304 * A176306 A176307 A176308

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 14 2010

Status

approved