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A141600
Triangle read by rows: T(n, k) = round( f(n)/(f(k)*f(n-k)) ), where f(n) = (n-1)*(n+1)!, and f(0) = f(1) = 1.
2
1, 1, 1, 1, 6, 1, 1, 8, 8, 1, 1, 8, 10, 8, 1, 1, 8, 10, 10, 8, 1, 1, 9, 12, 11, 12, 9, 1, 1, 10, 14, 14, 14, 14, 10, 1, 1, 10, 17, 18, 20, 18, 17, 10, 1, 1, 11, 20, 24, 28, 28, 24, 20, 11, 1, 1, 12, 24, 31, 40, 43, 40, 31, 24, 12, 1, 1, 13, 28, 39, 55, 66, 66, 55, 39, 28, 13, 1
OFFSET
0,5
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) = n^2 - 1, b(0) = b(1) = 1.
T(n, k) = round( f(n)/(f(k)*f(n-k)) ), where f(n) = (n-1)*(n+1)!, and f(0) = f(1) = 1.
T(n, n-k) = T(n, k).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 8, 8, 1;
1, 8, 10, 8, 1;
1, 8, 10, 10, 8, 1;
1, 9, 12, 11, 12, 9, 1;
1, 10, 14, 14, 14, 14, 10, 1;
1, 10, 17, 18, 20, 18, 17, 10, 1;
1, 11, 20, 24, 28, 28, 24, 20, 11, 1;
1, 12, 24, 31, 40, 43, 40, 31, 24, 12, 1;
MATHEMATICA
f[n_]:= If[n<2, 1, (n-1)*(n+1)!];
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 (n-1)*Factorial(n+1) >;
A141600:= func< n, k | Round(f(n)/(f(k)*f(n-k))) >;
[A141600(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 (n-1)*factorial(n+1)
def A141600(n, k): return round(f(n)/(f(k)*f(n-k)))
flatten([[A141600(n, k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Sep 20 2024
CROSSREFS
Sequence in context: A171147 A171695 A179233 * A303489 A195408 A011491
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited and new name by G. C. Greubel, Sep 20 2024
STATUS
approved