A001820 Central factorial numbers: 2nd subdiagonal of A008955.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A049033.

Third right-hand column of triangle A008955.

Programs

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 *)

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