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A098361
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Multiplication table of the factorial numbers read by antidiagonals.
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8
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1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 24, 120, 24, 12, 12, 24, 120, 720, 120, 48, 36, 48, 120, 720, 5040, 720, 240, 144, 144, 240, 720, 5040, 40320, 5040, 1440, 720, 576, 720, 1440, 5040, 40320, 362880, 40320, 10080, 4320, 2880, 2880, 4320, 10080, 40320, 362880
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - Stephen Crowley, Mar 28 2007
The triangle T(n, k) = k!*(n-k)! appears as denominators in the coefficients of the Niven polynomials x^n*(1 - x)^n/n! = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!). These polynomials are used in a proof that Pi^2 (hence Pi) is irrational. See the Niven and Havil references. - Wolfdieter Lang, May 07 2018; corrected by Dimitri Papadopoulos, Nov 30 2023
The case T(n+1,k) = k!*(n-k+1)!, 1 <= k <= n+1, n >= 0 is the number of choices for forming a cluster (compact group) of k numbered items arranged in a line on a set of permutations of n numbered items arranged in a line. - Igor Victorovich Statsenko, Oct 13 2023
The numbers T(n,k) also appear in the denominators of the partial fraction expansion of 1/(x*(x+1)*...*(x+n)) = Sum_{k=0..n} (-1)^k * 1/(T(n,k)*(x+k)). - Dimitri Papadopoulos, Nov 30 2023
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REFERENCES
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R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.
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LINKS
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FORMULA
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T(n, k) = k!*(n-k)! = n!/C(n,k), (0<=k<=n). - Peter Luschny, Aug 23 2010
Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - R. J. Mathar, Dec 10 2010
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EXAMPLE
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The array A(n, k) starts in row n=0 with columns k >= 0 as:
1, 1, 2, 6, 24, 120, ...
1, 1, 2, 6, 24, 120, ...
2, 2, 4, 12, 48, 240, ...
6, 6, 12, 36, 144, 720, ...
24, 24, 48, 144, 576, 2880, ...
120, 120, 240, 720, 2880, 14400, ...
720, 720, 1440, 4320, 17280, 86400, ...
5040, 5040, 10080, 30240, 120960, 604800, ...
40320, 40320, 80640, 241920, 967680, 4838400, ...
362880, 362880, 725760, 2177280, 8709120, 43545600, ...
...
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10...
0: 1
1: 1 1
2: 2 1 2
3: 6 2 2 6
4: 24 6 4 6 24
5: 120 24 12 12 24 120
6: 720 120 48 36 48 120 720
7: 5040 720 240 144 144 240 720 5040
8: 40320 5040 1440 720 576 720 1440 5040 40320
9: 362880 40320 10080 4320 2880 2880 4320 10080 40320 362880
10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800
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MAPLE
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seq(print(seq(k!*(n-k)!, k=0..n)), n=0..6); # Peter Luschny, Aug 23 2010
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MATHEMATICA
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Table[(n+1)!*Beta[n-k+1, k+1], {n, 0, 12}, {k, 0, n}]//Flatten (* Roger L. Bagula, Oct 29 2008 *)
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PROG
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(Magma) F:=Factorial; [F(n-k)*F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2022
(SageMath) f=factorial; flatten([[f(n-k)*f(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 12 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004
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STATUS
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approved
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