A001821 Central factorial numbers. (Formerly M5215 N2269)

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

Offset

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Fourth right-hand column of triangle A008955.

Sequence in context: A180812 A292002 A160313 * A027488 A269541 A280216

Adjacent sequences: A001818 A001819 A001820 * A001822 A001823 A001824

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Ralf Stephan, Aug 22 2004

Status

approved