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

1,5

Links

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

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)

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A000108, A174696, A176013.

Sequence in context: A176492 A174731 A342891 * A353952 A340432 A156539

Adjacent sequences: A174691 A174692 A174693 * A174695 A174696 A174697

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 27 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved