Search: id:A001791
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%I A001791 M3500 N1421
%S A001791 0,1,4,15,56,210,792,3003,11440,43758,167960,646646,2496144,9657700,
%T A001791 37442160,145422675,565722720,2203961430,8597496600,33578000610,
%U A001791 131282408400,513791607420,2012616400080,7890371113950,30957699535776
%N A001791 Binomial coefficients C(2n,n-1).
%C A001791 Number of peaks at even level in all Dyck paths of semilength n+1. Example: 
               a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where 
               U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A001791 Also number of long ascents (i.e. ascents of length at least two) in 
               all Dyck paths of semilength n+1. Example: a(2)=4 because in the 
               five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, 
               (UU)DUDD and (UUU)DDD, we have four long ascents (shown between parentheses). 
               Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e. vertices 
               of outdegree at least two) in all ordered trees with n+1 edges. - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004
%C A001791 Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,
               1) which touch or cross the line x-y=1. Example: For n=2 these are 
               the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba (kociemba(AT)t-online.de), 
               May 23 2004
%C A001791 Narayana transform (A001263) of [1, 3, 5, 7, 9,...] = (1, 4, 15, 56, 
               210,...). Row sums of triangles A136534 and A136536. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jan 04 2008
%C A001791 Starting with offset 1 = the Catalan sequence starting (1, 2, 5, 14,...) 
               convolved with A000984: (1, 2, 6, 20,...). [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), May 17 2009]
%D A001791 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001791 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001791 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 828.
%D A001791 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, 
               p. 517.
%D A001791 R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost 
               cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference 
               on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited 
               R. C. Mullin et al., 1970.
%D A001791 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A001791 T. D. Noe, <a href="b001791.txt">Table of n, a(n) for n=0..200</a>
%H A001791 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A001791 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%F A001791 G.f.: x*diff(c(x), x), c(x) = Catalan g.f. (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
%F A001791 Convolution of A001700( central binomial of odd order) and A000108 (Catalan): 
               a(n+1)=sum(C(k)*binomial(2*(n-k)+1, n-k), k=0..n), C(k): Catalan 
               [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ].
%F A001791 E.g.f.: exp(2x) I_1(2x), where I_1 is Bessel function. - Michael Somos, 
               Sep 08 2002
%F A001791 a(n)=sum{k=0..n, C(n, k)C(n, k+1) } - Paul Barry (pbarry(AT)wit.ie), 
               May 15 2003
%F A001791 a(n)=sum(binomial(i+n-1, n), i=1..n).
%F A001791 G.f.=[1-2x-sqrt(1-4x)]/[2xsqrt(1-4x)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 05 2003
%F A001791 A092956/(n!) - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 16 
               2004
%F A001791 a(n)=C(2n, n)-C(n); - Paul Barry (pbarry(AT)wit.ie), Apr 21 2005
%F A001791 a(n)=(1/(2*pi))*Int(x^n*(x-2)/sqrt(x(4-x)),x,0,4) is the moment sequence 
               representation; - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
%F A001791 Row sums of triangle A132812 starting (1, 4, 15, 56, 210,...). - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2007
%F A001791 Starting (1, 4, 15, 56, 210,...) gives the binomial transform of A025566 
               starting (1, 3, 8, 22, 61, 171,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 01 2007
%p A001791 [seq(binomial(2*n,n)/(n+1)*n,n=0..30)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2006
%p A001791 a:=n->add(binomial(2*n,n)/(n+1), k=1..n): seq(a(n), n=0..24); - ZerinvaryLajos 
               (zerinvarylajos(AT)yahoo.com), Oct 02 2007
%o A001791 (PARI) a(n)=if(n<1,0,(2*n)!/(n+1)!/(n-1)!)
%o A001791 (Mupad) combinat::catalan(n) *binomial(n,1) $ n = 0..24 - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Feb 15 2007
%Y A001791 n*C(n), C(n)=Catalan A000108. Cf. A000984.
%Y A001791 Diagonal 3 of triangle A100257.
%Y A001791 First differences are in A076540.
%Y A001791 Cf. A000108, A000984, A002378.
%Y A001791 Cf. A025566, A132812.
%Y A001791 Sequence in context: A026030 A047038 A158500 this_sequence A047128 A087438 
               A131497
%Y A001791 Adjacent sequences: A001788 A001789 A001790 this_sequence A001792 A001793 
               A001794
%K A001791 nonn,easy,nice
%O A001791 0,3
%A A001791 N. J. A. Sloane (njas(AT)research.att.com).
%E A001791 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000
%E A001791 More terms from ZerinvaryLajos (zerinvarylajos(AT)yahoo.com), Oct 02 
               2007

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