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A002453
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Central factorial numbers.
(Formerly M5249 N2283)
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4
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1, 35, 966, 24970, 631631, 15857205, 397027996, 9931080740, 248325446061, 6208571999575, 155218222621826, 3880490869237710, 97012589464171291, 2425317596203339145, 60632965641474990456, 1515824372664398367880
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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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).
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FORMULA
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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(0) = 1, a(1) = 35, a(2) = 966, a(n) = 35*a(n-1) -259*a(n-2) +225*a(n-3). - Harvey P. Dale, Feb 25 2015
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MAPLE
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A002453:=-1/(z-1)/(25*z-1)/(9*z-1); # Simon Plouffe (from his 1992 dissertation).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Right-hand column 2 in triangle A008958.
Sequence in context: A080250 A014934 A115473 * A240826 A210313 A049395
Adjacent sequences: A002450 A002451 A002452 * A002454 A002455 A002456
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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