A141601 Triangle read by rows: T(n, k) = round(f(n)/(f(k)*f(n-k))), where f(n) = n!*binomial(n,2), f(0) = f(1) = 1.

1, 1, 1, 1, 2, 1, 1, 9, 9, 1, 1, 8, 36, 8, 1, 1, 8, 33, 33, 8, 1, 1, 9, 38, 33, 38, 9, 1, 1, 10, 44, 41, 41, 44, 10, 1, 1, 11, 52, 52, 54, 52, 52, 11, 1, 1, 12, 62, 67, 76, 76, 67, 62, 12, 1, 1, 12, 72, 86, 105, 113, 105, 86, 72, 12, 1, 1, 13, 84, 108, 144, 169, 169, 144, 108, 84, 13, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = round( f(n)/(f(k)*f(n-k)) ), where f(n) = n*b(n)*f(n-1)/b(n-1), f(0) = f(1) = 1, b(n) = binomial(n, 2), b(0) = b(1) = 1.

Example

Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  9,  9,  1;
  1,  8, 36,  8,   1;
  1,  8, 33, 33,   8,   1;
  1,  9, 38, 33,  38,   9,   1;
  1, 10, 44, 41,  41,  44,  10,  1;
  1, 11, 52, 52,  54,  52,  52, 11,  1;
  1, 12, 62, 67,  76,  76,  67, 62, 12,  1;
  1, 12, 72, 86, 105, 113, 105, 86, 72, 12,  1;
  ...

Mathematica

f[n_]:= If[n<2, 1, n!*Binomial[n, 2]];
T[n_, k_]:= Round[f[n]/(f[n-k]*f[k])];
Table[T[n, k], {n, 0, 14}, {k, 0, n}]//Flatten

Prog

(Magma)
f:= func< n | n le 1 select 1 else Factorial(n)*Binomial(n, 2) >;
A141601:= func< n, k | Round(f(n)/(f(k)*f(n-k))) >;
[A141601(n, k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 20 2024
(SageMath)
def f(n): return 1 if (n<2) else factorial(n)*binomial(n, 2)
def A141601(n, k): return round(f(n)/(f(k)*f(n-k)))
flatten([[A141601(n, k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Sep 20 2024

Crossrefs

Cf. A141600.

Sequence in context: A214506 A246664 A229962 * A108558 A128434 A176417

Adjacent sequences: A141598 A141599 A141600 * A141602 A141603 A141604

Keyword

nonn,easy,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

Extensions

Edited, and new name, by G. C. Greubel, Sep 20 2024

Status

approved