A098361 Multiplication table of the factorial numbers read by antidiagonals.

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

Offset: 0

Author

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

Keywords

• nonn
• tabl

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; 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

It follows from the previous comment that the numbers T(n,k) also appear in the denominators of the coefficients of the logarithms of the integral of 1/(x*(x+1)*...*(x+n)): c + Sum{k=0...n} (-1)^k * 1/(T(n,k)) * ln(x+k). - Colin Linzer, Dec 18 2024

Examples

			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
		

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

• Stefano Spezia, First 101 antidiagonals of the array, flattened
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• I. V. Statsenko, Problem on variants of cluster formation at permutations in ordered structures, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2023, pp. 7-10. In Russian.

Crossrefs

Row sums A003149.

Cf. A003991, A098358, A098359, A098360.

Programs

Magma

F:=Factorial; [F(n-k)*F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2022

Maple

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

Mathematica

Table[(n+1)!*Beta[n-k+1, k+1], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Oct 29 2008 *)

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

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

E.g.f. as array: 1/((1 - x)*(1 - y)). - Stefano Spezia, Jul 10 2020