A001820 - OEIS

%I M4952 N2121 #45 Jan 13 2025 10:44:00

%S 1,14,273,7645,296296,15291640,1017067024,84865562640,8689315795776,

%T 1071814846360896,156823829909121024,26862299458337581056,

%U 5325923338791614078976,1210310405427816646041600,312542036038910895995289600,91018216923341770801874534400

%N Central factorial numbers: 2nd subdiagonal of A008955.

%C a(n-2) is the coefficient of x^3 in Product_{k=0..n} (x + k^2).

%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="/A001820/b001820.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+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

%F 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

%F a(n) ~ Pi^5 * n^(2*n+5) / (60 * exp(2*n)). - _Vaclav Kotesovec_, Feb 23 2015

%p 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

%t 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 *)

%Y Cf. A049033.

%Y Third right-hand column of triangle A008955.

%K nonn,changed

%O 0,2

%A _N. J. A. Sloane_