|
| |
|
|
A000346
|
|
2^(2*n+1) - binomial(2*n+1,n+1).
(Formerly M3920 N1611)
|
|
26
|
|
|
|
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;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Also a(n) = 2nd elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).
Also 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, Jul 14 2006
Hankel transform is (-1)^n*(2n+1); the Hankel transform of sum(k=0..n, C(2*n,k) ) - C(2n,n) is (-1)^n*n. - Paul Barry, Jan 21 2007
Row sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2) (A187926). [Emanuele Munarini, Mar 16 2011]
|
|
|
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.
M. Merca, A Special Case of the Generalized Girard-Waring Formula, Journal of Integer Sequences, Vol. 15 (2012), #12.5.7. - From N. J. A. Sloane, Nov 25 2012
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).
T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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).
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
|
Vincenzo Librandi, Table of n, a(n) for n = 0..500
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 185
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
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). - Philippe Deléham, Jan 22 2004
a(n)=sum{k=0..n+1, binomial(n+k, k-1)2^(n-k+1)} - Paul Barry, 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, Jan 21 2007
Logarithm g.f. log(1/(2-C(x)))=sum(n>0, a(n)/n*x^n), C(x)=(1-sqrt(1-4*x))/2x (A000108) [From Vladimir Kruchinin, Aug 10 2010]
Recurrence: (n+3) a(n+2) - 2(4n+9) a(n+1) + 8(2n+3) a(n) = 0. [Emanuele Munarini, Mar 16 2011]
E.g.f.:exp(2*x)*(2*exp(2*x) - BesselI(0,2*x) - BesselI(1,2*x)).
This is the first derivative of exp(2*x)*(exp(2*x) - BesselI(0,2*x))/2. See the e.g.f. of A032443 (which has a plus sign) and the remarks given there. [From Wolfdieter Lang, Jan 16 2012]
a(n)=sum_{0<=i<j<=n+1} binomial(n+1,i)*binomial(n+1,j) [From Mircea Merca, Apr 05 2012]
|
|
|
MAPLE
|
seq(2^(2*n+1)-binomial(2*n, n)*(2*n+1)/(n+1), n=0..12); [Emanuele Munarini, Mar 16 2011]
|
|
|
MATHEMATICA
|
Table[2^(2n+1)-Binomial[2n, n](2n+1)/(n+1), {n, 0, 20}] [Emanuele Munarini, Mar 16 2011]
|
|
|
PROG
|
(PARI) {a(n)=if(n<0, 0, 2^(2*n+1) -binomial(2*n+1, n))} /* Michael Somos, Oct 31 2006 */
(Maxima) makelist(2^(2*n+1)-binomial(2*n, n)*(2*n+1)/(n+1), n, 0, 12); [Emanuele Munarini, Mar 16 2011]
(MAGMA) [2^(2*n+1) - Binomial(2*n+1, n+1): n in [0..30]]; // Vincenzo Librandi, Jun 07 2011
|
|
|
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).
Sequence in context: A010036 A127617 A095932 * A026672 A049652 A026877
Adjacent sequences: A000343 A000344 A000345 * A000347 A000348 A000349
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane, Wolfdieter Lang
|
|
|
EXTENSIONS
|
Corrected by Christian G. Bower
|
|
|
STATUS
|
approved
|
| |
|
|
