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A098361 Multiplication table of the factorial numbers read by antidiagonals. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - Stephen Crowley, Mar 28 2007

Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - Roger L. Bagula, Oct 29 2008

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..53.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

FORMULA

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

EXAMPLE

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

... - Wolfdieter Lang, May 07 2018.

MAPLE

seq(print(seq(k!*(n-k)!, k=0..n)), n=0..6); # Peter Luschny, Aug 23 2010

MATHEMATICA

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

CROSSREFS

Row sums A003149.

Cf. A003991, A098358, A098359, A098360.

Sequence in context: A152431 A143965 A182073 * A050977 A053448 A060550

Adjacent sequences:  A098358 A098359 A098360 * A098362 A098363 A098364

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified September 18 12:45 EDT 2019. Contains 327170 sequences. (Running on oeis4.)