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A176305 Triangle T(n,k) = 1 -A002627(k) -A002627(n-k) +A002627(n), read by rows. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 16 14:39 EDT 2021. Contains 343949 sequences. (Running on oeis4.)