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A000597 Central factorial numbers.
(Formerly M5255 N2287)
5

%I M5255 N2287 #105 Jul 11 2023 12:21:43

%S 36,820,7645,44473,191620,669188,1999370,5293970,12728936,28285400,

%T 58856655,115842675,217378200,391367064,679524340,1142659012,

%U 1867463260,2975110060,4631998657,7063027565,10567817084,15540347900,22492529150,32082258390,45146587200

%N Central factorial numbers.

%C a(n) is the sum of all products of three distinct squares of positive integers up to n, i.e., the sum of all products of three distinct elements from the set of squares {1^2, ..., (n-1)^2}. - _Roudy El Haddad_, Feb 17 2022

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A000597/b000597.txt">Table of n, a(n) for n = 4..1000</a>

%H Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.

%H Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.200-233">A generalization of multiple zeta value. Part 2: Multiple sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233. (See Example 5.2 and Theorem 5.1)

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F O.g.f.: x^4 * (x^5 + 75*x^4 + 603*x^3 + 1065*x^2 + 460*x + 36) / (1-x)^10.

%F a(n) = s(n,n-3)^2-2*s(n,n-4)*s(n,n-2)+2*s(n,n-5)*s(n,n-1)+2*s(n,n-6), where s(n,k) are Stirling numbers of the first kind, A048994. - _Mircea Merca_, Apr 03 2012

%F From _Roudy El Haddad_, Feb 17 2022: (Start)

%F a(n) = Sum_{0 < i < j < k < n} (i*j*k)^2.

%F a(n) = (n - 1)*(n - 2)*(n - 3)*n*(2*n-1)*(2*n - 3)*(2*n - 5)*(35*n^2 + 21*n + 4)/45360.

%F a(n) = (1/(9!*2))*((2*n)!/(2*n-7)!)*(35*n^2 + 21*n + 4).

%F a(n) = binomial(2*n,7)*(35*n^2 + 21*n + 4)/144. (End)

%p 1/(-1+z)^10*(z^5+75*z^4+603*z^3+1065*z^2+460*z+36);

%p seq(stirling1(n,n-3)^2-2*stirling1(n,n-4)*stirling1(n,n-2)+2*stirling1(n,n-5)*stirling1(n,n-1)+2*stirling1(n,n-6),n=0..30); # _Mircea Merca_, Apr 03 2012

%t CoefficientList[Series[(x^5 + 75*x^4 + 603*x^3 + 1065*x^2 + 460*x + 36)/(1-x)^10, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 23 2015 *)

%t LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {36, 820, 7645, 44473, 191620, 669188, 1999370, 5293970, 12728936, 28285400}, 40] (* _Vincenzo Librandi_, Aug 07 2017 *)

%o (PARI) {a(n) = (n-1)*(n-2)*(n-3)*(n)*(2*n-1)*(2*n-3)*(2*n-5)*(35*n^2+21*n+4)/45360}; \\ _Roudy El Haddad_, Feb 17 2022

%Y Column 3 of triangle A008955.

%Y Cf. A000290 (squares), A000330 (sum of squares), A000596 (for two squares).

%Y Cf. A001303 (for power 1).

%K nonn,easy

%O 4,1

%A _N. J. A. Sloane_

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)