This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A098361 #76 Dec 20 2024 23:53:22
%S A098361 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 A098361 120,720,5040,720,240,144,144,240,720,5040,40320,5040,1440,720,576,
%U A098361 720,1440,5040,40320,362880,40320,10080,4320,2880,2880,4320,10080,40320,362880
%N A098361 Multiplication table of the factorial numbers read by antidiagonals.
%C A098361 This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - _Stephen Crowley_, Mar 28 2007
%C A098361 Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - _Roger L. Bagula_, Oct 29 2008
%C A098361 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
%C A098361 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
%C A098361 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
%C A098361 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
%D A098361 R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
%D A098361 Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
%D A098361 Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.
%H A098361 Stefano Spezia, <a href="/A098361/b098361.txt">First 101 antidiagonals of the array, flattened</a>
%H A098361 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.
%H A098361 I. V. Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2023-10-1.pdf#page=7">Problem on variants of cluster formation at permutations in ordered structures</a>, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2023, pp. 7-10. In Russian.
%F A098361 T(n, k) = k!*(n-k)! = n!/C(n,k), (0<=k<=n). - _Peter Luschny_, Aug 23 2010
%F A098361 Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - _R. J. Mathar_, Dec 10 2010
%F A098361 E.g.f. as array: 1/((1 - x)*(1 - y)). - _Stefano Spezia_, Jul 10 2020
%e A098361 The array A(n, k) starts in row n=0 with columns k >= 0 as:
%e A098361 1, 1, 2, 6, 24, 120, ...
%e A098361 1, 1, 2, 6, 24, 120, ...
%e A098361 2, 2, 4, 12, 48, 240, ...
%e A098361 6, 6, 12, 36, 144, 720, ...
%e A098361 24, 24, 48, 144, 576, 2880, ...
%e A098361 120, 120, 240, 720, 2880, 14400, ...
%e A098361 720, 720, 1440, 4320, 17280, 86400, ...
%e A098361 5040, 5040, 10080, 30240, 120960, 604800, ...
%e A098361 40320, 40320, 80640, 241920, 967680, 4838400, ...
%e A098361 362880, 362880, 725760, 2177280, 8709120, 43545600, ...
%e A098361 ...
%e A098361 The triangle T(n, k) begins:
%e A098361 n\k 0 1 2 3 4 5 6 7 8 9 10...
%e A098361 0: 1
%e A098361 1: 1 1
%e A098361 2: 2 1 2
%e A098361 3: 6 2 2 6
%e A098361 4: 24 6 4 6 24
%e A098361 5: 120 24 12 12 24 120
%e A098361 6: 720 120 48 36 48 120 720
%e A098361 7: 5040 720 240 144 144 240 720 5040
%e A098361 8: 40320 5040 1440 720 576 720 1440 5040 40320
%e A098361 9: 362880 40320 10080 4320 2880 2880 4320 10080 40320 362880
%e A098361 10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800
%e A098361 ... - _Wolfdieter Lang_, May 07 2018
%p A098361 seq(print(seq(k!*(n-k)!,k=0..n)),n=0..6); # _Peter Luschny_, Aug 23 2010
%t A098361 Table[(n+1)!*Beta[n-k+1, k+1], {n,0,12}, {k,0,n}]//Flatten (* _Roger L. Bagula_, Oct 29 2008 *)
%o A098361 (Magma) F:=Factorial; [F(n-k)*F(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2022
%o A098361 (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
%Y A098361 Row sums A003149.
%Y A098361 Cf. A003991, A098358, A098359, A098360.
%K A098361 nonn,tabl
%O A098361 0,4
%A A098361 Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004