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Search: id:A000346
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| A000346 |
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2^{2n+1} - binomial(2n+1,n+1). Also 2nd elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).
(Formerly M3920 N1611)
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+0
21
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| 1, 5, 22, 93, 386, 1586, 6476, 26333, 106762, 431910, 1744436, 7036530, 28354132, 114159428, 459312152, 1846943453, 7423131482, 29822170718, 119766321572, 480832549478, 1929894318332, 7744043540348, 31067656725032
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n) = one half the sum of the heights, over all Dyck (n+2)-paths, of the vertices that are at even height and terminate an upstep. For example with n=1, these vertices are indicated by asterisks in the 5 Dyck 3-paths: UU*UDDD, UU*DU*DD, UDUU*DD, UDUDUD, UU*DDUD, yielding a(1)=(2+4+2+0+2)/2=5. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
Hankel transform is (-1)^n*(2n+1); the Hankel transform of sum{k=0..n, C(2n,k)}-C(2n,n) is (-1)^n*n. - Paul Barry (pbarry(AT)wit.ie), Jan 21 2007
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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).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=2).
W. T. Tutte, On the enumeration of planar maps. Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
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LINKS
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D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 185
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FORMULA
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G.f.: c(x)/(1-4x), c(x) = g.f. of Catalan numbers
Convolution of Catalan numbers and powers of 4.
Also one half of convolution of central binomial coeffs. A000984(n), n=0, 1, 2, ... with shifted central binomial coeffs. A000984(n), n=1, 2, 3, ...
a(n) = Sum_{k=0..n} A000984(k)*A001700(n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 22 2004
a(n)=sum{k=0..n+1, binomial(n+k, k-1)2^(n-k+1)} - Paul Barry (pbarry(AT)wit.ie), Nov 13 2004
a(n) = Sum_{i=0..n} binomial(2n+2, i). See A008949. - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
a(n)=sum{k=0..n+1, C(2n+2,k)}-C(2n+2,n+1); - Paul Barry (pbarry(AT)wit.ie), Jan 21 2007
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, 2^(2*n+1) -binomial(2*n+1, n))} /* Michael Somos Oct 31 2006 */
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CROSSREFS
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Cf. A000108, A014137, A014318. A column of A058893.
Bisection of A058622 and (possibly) A007008.
a(n)=A045621(2n+1) = (1/2)*A068551(n+1).
Sequence in context: A010036 A127617 A095932 this_sequence A026672 A049652 A026877
Adjacent sequences: A000343 A000344 A000345 this_sequence A000347 A000348 A000349
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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EXTENSIONS
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Corrected by Christian Bower (bowerc(AT)usa.net)
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