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A156045
Triangle T(n, k) = 2 + n! - k! - (n-k)!, read by rows.
1
1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 19, 22, 19, 1, 1, 97, 114, 114, 97, 1, 1, 601, 696, 710, 696, 601, 1, 1, 4321, 4920, 5012, 5012, 4920, 4321, 1, 1, 35281, 39600, 40196, 40274, 40196, 39600, 35281, 1, 1, 322561, 357840, 362156, 362738, 362738, 362156, 357840, 322561, 1
OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 4, 12, 62, 424, 3306, 28508, 270430, 2810592, 31840994, ...}.
FORMULA
T(n, k) = 2 + n! - k! - (n-k)!.
Sum_{k=0..n} T(n,k) = 2*(n+1) + (n+1)! - 2*!(n+1), where !n = A003422(n). - G. C. Greubel, Dec 02 2019
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 5, 5, 1;
1, 19, 22, 19, 1;
1, 97, 114, 114, 97, 1;
1, 601, 696, 710, 696, 601, 1;
1, 4321, 4920, 5012, 5012, 4920, 4321, 1;
1, 35281, 39600, 40196, 40274, 40196, 39600, 35281, 1;
MAPLE
seq(seq( n! -k! -(n-k)! +2, k=0..n), n=0..10); # G. C. Greubel, Dec 02 2019
MATHEMATICA
Table[n! -k! -(n-k)! +2, {n, 0, 10}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = n! -k! -(n-k)! +2; \\ G. C. Greubel, Dec 02 2019
(Magma) F:=Factorial; [F(n) -F(k) -F(n-k) +2: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 02 2019
(SageMath) f=factorial; [[f(n) -f(k) -f(n-k) +2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 02 2019
(GAP) F:=Factorial;; Flat(List([0..10], n-> List([0..n], k-> F(n) -F(k) -F(n-k) +2 ))); # G. C. Greubel, Dec 02 2019
CROSSREFS
Sequence in context: A378173 A060854 A091378 * A119687 A086856 A052916
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 02 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 02 2019
STATUS
approved