A001820 Central factorial numbers. (Formerly M4952 N2121)

1, 14, 273, 7645, 296296, 15291640, 1017067024, 84865562640, 8689315795776, 1071814846360896, 156823829909121024, 26862299458337581056, 5325923338791614078976, 1210310405427816646041600, 312542036038910895995289600, 91018216923341770801874534400

Offset

0,2

Comments

a(n-2) is the coefficient of x^3 in Product_{k=0..n} (x + k^2).

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

T. D. Noe, Table of n, a(n) for n = 0..100

Takao Komatsu, Convolution identities of poly-Cauchy numbers with level 2, arXiv:2003.12926 [math.NT], 2020.

Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

Formula

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

a(n) = (3*n^2 + 6*n + 5)*a(n-1) - (n^2 + n + 1)*(3*n^2 + 3*n + 1)*a(n-2) + n^6*a(n-3). - Vaclav Kotesovec, Feb 23 2015

a(n) ~ Pi^5 * n^(2*n+5) / (60 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Maple

seq(2*Stirling1(n+3, 1)*Stirling1(n+3, 5)-2*Stirling1(n+3, 2)*Stirling1(n+3, 4)+Stirling1(n+3, 3)^2, n=0..20); # Mircea Merca, Apr 03 2012

Mathematica

Table[StirlingS1[n+3, 3]^2 - 2*StirlingS1[n+3, 2]*StirlingS1[n+3, 4] + 2*StirlingS1[n+3, 1]*StirlingS1[n+3, 5], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

Crossrefs

Cf. A049033.

Third right-hand column of triangle A008955.

Sequence in context: A138560 A051690 A048668 * A211900 A215544 A205353

Adjacent sequences: A001817 A001818 A001819 * A001821 A001822 A001823

Keyword

nonn

Author

N. J. A. Sloane

Status

approved