A000597 Central factorial numbers. (Formerly M5255 N2287)

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

4,1

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).

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)

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 *)

Prog

(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

Crossrefs

Column 3 of triangle A008955.

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

Cf. A001303 (for power 1).

Sequence in context: A028222 A028216 A028221 * A028214 A028197 A028209

Adjacent sequences: A000594 A000595 A000596 * A000598 A000599 A000600

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved