A002453 Central factorial numbers. (Formerly M5249 N2283)

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

Offset

0,2

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).

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

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 *)

Prog

(GAP) List([0..20], n->(5^(2*n+4)-3^(2*n+5)+2)/384); # Muniru A Asiru, Dec 20 2018
(PARI) vector(20, n, n--; (5^(2*n+4)-3^(2*n+5)+2)/384) \\ G. C. Greubel, Jul 04 2019
(Magma) [(5^(2*n+4)-3^(2*n+5)+2)/384: n in [0..20]]; // 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

Crossrefs

Right-hand column 2 in triangle A008958.

Cf. A002452.

Sequence in context: A080250 A014934 A115473 * A240826 A210313 A049395

Adjacent sequences: A002450 A002451 A002452 * A002454 A002455 A002456

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved