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A003518
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a(n) = 8*binomial(2*n+1,n-3)/(n+5).
(Formerly M4529)
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26
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1, 8, 44, 208, 910, 3808, 15504, 62016, 245157, 961400, 3749460, 14567280, 56448210, 218349120, 843621600, 3257112960, 12570420330, 48507033744, 187187399448, 722477682080, 2789279908316, 10772391370048, 41620603020640
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OFFSET
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3,2
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COMMENTS
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a(n-6) is the number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+4,n-3). - Emeric Deutsch, May 30 2004
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
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FORMULA
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G.f.: x^3*C(x)^8, where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=7, a(n-4)=(-1)^(n-7)*coeff(charpoly(A,x),x^7). - Milan Janjic, Jul 08 2010
Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e., in Maple notation: a(n+3) = int(x^n*W(x),x=0..4), n=0,1..., with W(x) = (1/2)*x^(7/2)*(x-2)*(x^2-4*x+2)*sqrt(4-x)/Pi. - Karol A. Penson, Oct 26 2016
E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.
a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)
D-finite with recurrence: -(n+5)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
Sum_{n>=3} 1/a(n) = 43*Pi/(36*sqrt(3)) - 81/80.
Sum_{n>=3} (-1)^(n+1)/a(n) = 6213*log(phi)/(50*sqrt(5)) - 10339/400, where phi is the golden ratio (A001622). (End)
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EXAMPLE
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G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...
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MATHEMATICA
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Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* Michael De Vlieger, Oct 26 2016 *)
CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 23 2017 *)
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PROG
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(PARI) {a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* Michael Somos, Mar 14 2011 */
(PARI) x='x+O('x^50); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ Altug Alkan, Nov 01 2015
(Magma) [8*Binomial(2*n+1, n-3)/(n+5): n in [3..30]]; // Vincenzo Librandi, Jan 23 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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