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A098361 Multiplication table of the factorial numbers read by antidiagonals. 4

%I

%S 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,

%T 120,720,5040,720,240,144,144,240,720,5040,40320,5040,1440,720,576,

%U 720,1440,5040,40320,362880,40320,10080,4320,2880,2880,4320,10080,40320

%N Multiplication table of the factorial numbers read by antidiagonals.

%C This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - _Stephen Crowley_, Mar 28 2007

%C Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - _Roger L. Bagula_, Oct 29 2008

%C 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

%D R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.

%D Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.

%D Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%F T(n, k) = k!*(n-k)! = n!/C(n,k), (0<=k<=n). - _Peter Luschny_, Aug 23 2010

%F Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - _R. J. Mathar_, Dec 10 2010

%e The array A(n, k) starts in row n=0 with columns k >= 0 as:

%e 1, 1, 2, 6, 24, 120, ...

%e 1, 1, 2, 6, 24, 120, ...

%e 2, 2, 4, 12, 48, 240, ...

%e 6, 6, 12, 36, 144, 720, ...

%e 24, 24, 48, 144, 576, 2880, ...

%e 120, 120, 240, 720, 2880, 14400, ...

%e 720, 720, 1440, 4320, 17280, 86400, ...

%e 5040, 5040, 10080, 30240, 120960, 604800, ...

%e 40320, 40320, 80640, 241920, 967680, 4838400, ...

%e 362880, 362880, 725760, 2177280, 8709120, 43545600, ...

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10...

%e 0: 1

%e 1: 1 1

%e 2: 2 1 2

%e 3: 6 2 2 6

%e 4: 24 6 4 6 24

%e 5: 120 24 12 12 24 120

%e 6: 720 120 48 36 48 120 720

%e 7: 5040 720 240 144 144 240 720 5040

%e 8: 40320 5040 1440 720 576 720 1440 5040 40320

%e 9: 362880 40320 10080 4320 2880 2880 4320 10080 40320 362880

%e 10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800

%e ... - _Wolfdieter Lang_, May 07 2018.

%p seq(print(seq(k!*(n-k)!,k=0..n)),n=0..6); # _Peter Luschny_, Aug 23 2010

%t Table[Table[(n + 1)!*Beta[n - m + 1, m + 1], {m, 0, n}], {n, 0, 10}] Flatten[%] (* _Roger L. Bagula_, Oct 29 2008 *)

%Y Row sums A003149.

%Y Cf. A003991, A098358, A098359, A098360.

%K nonn,tabl

%O 0,4

%A Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

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Last modified October 18 08:08 EDT 2019. Contains 328146 sequences. (Running on oeis4.)