A002453 Central factorial numbers: 2nd subdiagonal of A008958.

1, 35, 966, 24970, 631631, 15857205, 397027996, 9931080740, 248325446061, 6208571999575, 155218222621826, 3880490869237710, 97012589464171291, 2425317596203339145, 60632965641474990456, 1515824372664398367880

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

• 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.
• 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).
• T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..710
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for sequences related to factorial numbers
• Index entries for linear recurrences with constant coefficients, signature (35,-259,225).

Crossrefs

Right-hand column 2 in triangle A008958.

Cf. A002452.

Programs

GAP

List([0..20],n->(5^(2*n+4)-3^(2*n+5)+2)/384); # Muniru A Asiru, Dec 20 2018

Magma

[(5^(2*n+4)-3^(2*n+5)+2)/384: n in [0..20]]; // G. C. Greubel, Jul 04 2019

Maple

A002453:=-1/(z-1)/(25*z-1)/(9*z-1); # Simon Plouffe (from his 1992 dissertation).

Mathematica

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 *)

PARI

vector(20, n, n--; (5^(2*n+4)-3^(2*n+5)+2)/384) \\ G. C. Greubel, Jul 04 2019

Sage

[(5^(2*n+4)-3^(2*n+5)+2)/384 for n in (0..20)] # G. C. Greubel, Jul 04 2019

Formula

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

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

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

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

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

a(n) = 25*a(n-1) + A002452(n+1), with a(0) = 1. - Nadia Lafreniere, Aug 08 2022