A002453 - OEIS

%I M5249 N2283 #81 Jan 13 2025 10:51:11

%S 1,35,966,24970,631631,15857205,397027996,9931080740,248325446061,

%T 6208571999575,155218222621826,3880490869237710,97012589464171291,

%U 2425317596203339145,60632965641474990456,1515824372664398367880

%N Central factorial numbers: 2nd subdiagonal of A008958.

%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.

%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).

%D T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36.

%H G. C. Greubel, <a href="/A002453/b002453.txt">Table of n, a(n) for n = 0..710</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-259,225).

%F G.f.: 1/((1 - x)*(1 - 9*x)*(1 - 25*x)).

%F a(n) = (5^(2*n + 4) - 3^(2*n + 5) + 2)/384.

%F E.g.f.: sinh(x)^5/120 = Sum_{n>=0} a(n)*x^(2*n + 5)/(2*n + 5)!. - _Vladimir Kruchinin_, Sep 30 2012

%F a(n) = det(|v(i+3,j+2)|, 1 <= i,j <= n), where v(n,k) are central factorial numbers of the first kind with odd indices (A008956). - _Mircea Merca_, Apr 06 2013

%F a(n) = 35*a(n-1) -259*a(n-2) +225*a(n-3), with a(0) = 1, a(1) = 35, a(2) = 966. - _Harvey P. Dale_, Feb 25 2015

%F a(n) = 25*a(n-1) + A002452(n+1), with a(0) = 1. - _Nadia Lafreniere_, Aug 08 2022

%p A002453:=-1/(z-1)/(25*z-1)/(9*z-1); # _Simon Plouffe_ (from his 1992 dissertation).

%t CoefficientList[Series[1/((1-x)(1-9x)(1-25x)),{x,0,20}],x] (* or *) LinearRecurrence[{35,-259,225},{1,35,966},20] (* _Harvey P. Dale_, Feb 25 2015 *)

%o (GAP) List([0..20],n->(5^(2*n+4)-3^(2*n+5)+2)/384); # _Muniru A Asiru_, Dec 20 2018

%o (PARI) vector(20, n, n--; (5^(2*n+4)-3^(2*n+5)+2)/384) \\ _G. C. Greubel_, Jul 04 2019

%o (Magma) [(5^(2*n+4)-3^(2*n+5)+2)/384: n in [0..20]]; // _G. C. Greubel_, Jul 04 2019

%o (Sage) [(5^(2*n+4)-3^(2*n+5)+2)/384 for n in (0..20)] # _G. C. Greubel_, Jul 04 2019

%Y Right-hand column 2 in triangle A008958.

%Y Cf. A002452.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_