A174694 Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.

1, 1, 1, 1, 13, 1, 1, 121, 121, 1, 1, 1081, 2281, 1081, 1, 1, 10081, 35281, 35281, 10081, 1, 1, 100801, 524161, 876961, 524161, 100801, 1, 1, 1088641, 7862401, 19716481, 19716481, 7862401, 1088641, 1, 1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1

Offset: 1

Author

Roger L. Bagula, Mar 27 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begin as:
  1;
  1,        1;
  1,       13,         1;
  1,      121,       121,         1;
  1,     1081,      2281,      1081,         1;
  1,    10081,     35281,     35281,     10081,         1;
  1,   100801,    524161,    876961,    524161,    100801,         1;
  1,  1088641,   7862401,  19716481,  19716481,   7862401,   1088641,        1;
  1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000108, A174696, A176013.

Programs

Magma

A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;
[A174694(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021

Mathematica

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

Sage

def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1
flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

Formula

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

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

T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.

Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021