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A003518
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8*C(2*n+1,n-3)/(n+5).
(Formerly M4529)
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25
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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) = number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. Zoran Sunik reference) - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+4,n-3). - Emeric Deutsch, May 30 2004
a(n) = A214292(2*n,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
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REFERENCES
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S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Shapiro, L. W.; A Catalan triangle. Discrete Math. 14 (1976), no. 1, 83-90.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.
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LINKS
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Table of n, a(n) for n=3..25.
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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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
The convolution of A002057 with itself. - Gerald McGarvey, Nov 08 2007
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). [From Milan Janjic, Jul 08 2010]
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EXAMPLE
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x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...
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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 */
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CROSSREFS
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Cf. A002057.
First differences are in A026018.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003519 A001392.
Sequence in context: A169795 A073380 A022636 * A100575 A003220 A125318
Adjacent sequences: A003515 A003516 A003517 * A003519 A003520 A003521
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Jon E. Schoenfield (jonscho(AT)hiwaay.net), May 06 2010
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STATUS
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approved
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