|
Search: id:A000346
|
|
|
| A000346 |
|
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)
|
|
+0
20
|
|
| 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)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
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
|
|
REFERENCES
|
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.
|
|
LINKS
|
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
|
|
FORMULA
|
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
|
|
PROGRAM
|
(PARI) {a(n)=if(n<0, 0, 2^(2*n+1) -binomial(2*n+1, n))} /* Michael Somos Oct 31 2006 */
|
|
CROSSREFS
|
Cf. A000108, A014137, A014318. A column of A058893.
Bisection of A058622 and (possibly) A007008.
a(n)=A045621(2n+1) = (1/2)*A068551(n+1).
Adjacent sequences: A000343 A000344 A000345 this_sequence A000347 A000348 A000349
Sequence in context: A010036 A127617 A095932 this_sequence A026672 A049652 A026877
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
njas, Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
|
|
EXTENSIONS
|
Corrected by Christian Bower (bowerc(AT)usa.net)
|
|
|
Search completed in 0.003 seconds
|
