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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Ralf Stephan, Aug 22 2004
STATUS
approved