A001821 - OEIS

%I M5215 N2269 #43 Jan 13 2025 10:43:02

%S 1,30,1023,44473,2475473,173721912,15088541896,1593719752240,

%T 201529405816816,30092049283982400,5242380158902146624,

%U 1054368810603158319360,242558905724502235934976,63305390270900389045395456,18607799824329123330114576384

%N Central factorial numbers: 3rd subdiagonal of A008955.

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

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001821/b001821.txt">Table of n, a(n) for n = 0..100</a>

%H Takao Komatsu, <a href="https://arxiv.org/abs/2003.12926">Convolution identities of poly-Cauchy numbers with level 2</a>, arXiv:2003.12926 [math.NT], 2020.

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

%F 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

%F 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

%F a(n) ~ Pi^7 * n^(2*n+7) / (2520 * exp(2*n)). - _Vaclav Kotesovec_, Feb 23 2015

%p 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

%t 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 *)

%o (Python)

%o from sympy.functions.combinatorial.numbers import stirling

%o def s(n, k): return stirling(n, k, kind=1)

%o 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)

%o print([a(n) for n in range(15)]) # _Michael S. Branicky_, Jan 30 2021

%Y Fourth right-hand column of triangle A008955.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Ralf Stephan_, Aug 22 2004