A174690 Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 13, 13, 1, 1, 73, 121, 73, 1, 1, 481, 1081, 1081, 481, 1, 1, 3601, 10081, 13681, 10081, 3601, 1, 1, 30241, 100801, 171361, 171361, 100801, 30241, 1, 1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1, 1, 2903041, 12700801, 30119041, 45360001, 45360001, 30119041, 12700801, 2903041, 1

Offset

0,5

Links

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

Formula

T(n, k) = n!*binomial(n, k) - n! + 1.

From G. C. Greubel, Feb 09 2021: (Start)

T(n, k) = A196347(n, k) - n! + 1 = (-1)^k * A021012(n, k) - n! + 1.

Sum_{k=0..n} T(n, k) = 2^n * n! - (n+1)! + (n+1) = A000165(n) - (n+1)! + (n+1). (End)

Example

Triangle begins as:
  1;
  1,      1;
  1,      3,       1;
  1,     13,      13,       1;
  1,     73,     121,      73,       1;
  1,    481,    1081,    1081,     481,       1;
  1,   3601,   10081,   13681,   10081,    3601,       1;
  1,  30241,  100801,  171361,  171361,  100801,   30241,      1;
  1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1;

Mathematica

T[n_, k_]:= n!*Binomial[n, k] - n! + 1;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage) flatten([[factorial(n)*(binomial(n, k) -1) + 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021
(Magma) [Factorial(n)*(Binomial(n, k) -1) + 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

Crossrefs

Cf. A000165, A021012, A196347.

Sequence in context: A055154 A338875 A015112 * A156869 A153090 A203002

Adjacent sequences: A174687 A174688 A174689 * A174691 A174692 A174693

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 27 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved