A000597 Central factorial numbers: A008955(n,3).

36, 820, 7645, 44473, 191620, 669188, 1999370, 5293970, 12728936, 28285400, 58856655, 115842675, 217378200, 391367064, 679524340, 1142659012, 1867463260, 2975110060, 4631998657, 7063027565, 10567817084, 15540347900, 22492529150, 32082258390, 45146587200

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 4..1000
• Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
• Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. 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)
• Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
• Index entries for sequences related to factorial numbers
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Column 3 of triangle A008955.

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

Cf. A001303 (for power 1).

Programs

Maple

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

Mathematica

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 *)
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 *)

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

Formula

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

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

From Roudy El Haddad, Feb 17 2022: (Start)

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

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.

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

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