A001821 Central factorial numbers: 3rd subdiagonal of A008955.

1, 30, 1023, 44473, 2475473, 173721912, 15088541896, 1593719752240, 201529405816816, 30092049283982400, 5242380158902146624, 1054368810603158319360, 242558905724502235934976, 63305390270900389045395456, 18607799824329123330114576384

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n-3) is the coefficient of x^4 in Product_{k=0..n} (x + k^2). - Ralf Stephan, Aug 22 2004

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

Fourth right-hand column of triangle A008955.

Programs

Maple

seq(Stirling1(n+4, 4)^2-2*Stirling1(n+4, 1)*Stirling1(n+4, 7)+2*Stirling1(n+4, 2)*Stirling1(n+4, 6) -2*Stirling1(n+4, 3)*Stirling1(n+4, 5), n=0..20); # Mircea Merca, Apr 03 2012

Mathematica

Table[StirlingS1[n+4, 4]^2 - 2*StirlingS1[n+4, 1]*StirlingS1[n+4, 7] + 2*StirlingS1[n+4, 2]*StirlingS1[n+4, 6] - 2*StirlingS1[n+4, 3]*StirlingS1[n+4, 5], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

Python

from sympy.functions.combinatorial.numbers import stirling
def s(n, k): return stirling(n, k, kind=1)
def a(n): return s(n+4, 4)**2 - 2*s(n+4, 1)*s(n+4, 7) + 2*s(n+4, 2)*s(n+4, 6) - 2*s(n+4, 3)*s(n+4, 5)
print([a(n) for n in range(15)]) # Michael S. Branicky, Jan 30 2021

Formula

a(n) = s(n+4,4)^2 - 2*s(n+4,1)*s(n+4,7) + 2*s(n+4,2)*s(n+4,6) - 2*s(n+4,3)*s(n+4,5), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012

a(n) = 2*(2*n^2 + 6*n + 7)*a(n-1) - 3*(2*n^4 + 8*n^3 + 16*n^2 + 16*n + 7)*a(n-2) + (2*n^2 + 2*n + 1)*(2*n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*a(n-3) - n^8*a(n-4). - Vaclav Kotesovec, Feb 23 2015

a(n) ~ Pi^7 * n^(2*n+7) / (2520 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Extensions

More terms from Ralf Stephan, Aug 22 2004