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 A000346 a(n) = 2^(2*n+1) - binomial(2*n+1, n+1). (Formerly M3920 N1611) 73
 1, 5, 22, 93, 386, 1586, 6476, 26333, 106762, 431910, 1744436, 7036530, 28354132, 114159428, 459312152, 1846943453, 7423131482, 29822170718, 119766321572, 480832549478, 1929894318332, 7744043540348, 31067656725032, 124613686513778, 499744650202436 (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 From Gus Wiseman, Jul 19 2021: (Start) For n > 0, a(n-1) is also the number of integer compositions of 2n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A053754 /\ A345921. For example, the a(3-1) = 22 compositions of 6 are:   (6)  (1,5)  (1,1,4)  (1,1,1,3)  (1,1,1,1,2)        (2,4)  (1,2,3)  (1,1,3,1)  (1,1,2,1,1)        (4,2)  (1,4,1)  (1,2,1,2)  (2,1,1,1,1)        (5,1)  (2,1,3)  (1,3,1,1)               (2,2,2)  (2,1,2,1)               (3,1,2)  (3,1,1,1)               (3,2,1)               (4,1,1) (End) REFERENCES 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..500 R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004. Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020. 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. E. Davenport, L. K. Pudwell, L. W. Shapiro and L. C. Woodson, The Boundary of Ordered Trees, 2014. P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 185 R. K. Guy, Letter to N. J. A. Sloane Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7. - From N. J. A. Sloane, Nov 25 2012 D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001. 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). John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers. 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 (eq. 5, b=1). N. J. A. Sloane, Notes 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) = A045621(2n+1) = (1/2)*A068551(n+1). 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). - Vladimir Kruchinin, Aug 10 2010 D-finite with 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. - Wolfdieter Lang, Jan 16 2012 a(n) = Sum_{0<=i-5. - Michael Somos, Jan 25 2014 REVERT transform is [1,-5,28,-168,1056,...] = alternating signed version of A069731. - Michael Somos, Jan 25 2014 Convolution square is A038806. - Michael Somos, Jan 25 2014 BINOMIAL transform of A055217(n-1) is a(n-1). - Michael Somos, Jan 25 2014 (n+1) * a(n) = A033504(n). - Michael Somos, Jan 25 2014 Recurrence: (n+1)*a(n) = 512*(2*n-7)*a(n-5) + 256*(13-5*n)*a(n-4) + 64*(10*n-17)*a(n-3) + 32*(4-5*n)*a(n-2) + 2*(10*n+1)*a(n-1), n>=5. - Fung Lam, Mar 21 2014 Asymptotic approximation: a(n) ~ 2^(2n+1)*(1-1/sqrt(n*Pi)). - Fung Lam, Mar 21 2014 a(n) = Sum_{m = n+2..2*(n+1)} binomial(2*(n+1), m), n >= 0. - Wolfdieter Lang, May 22 2015 a(n) = A000302(n) + A008549(n). - Gus Wiseman, Jul 19 2021 EXAMPLE G.f. = 1 + 5*x + 22*x^2 + 93*x^3 + 386*x^4 + 1586*x^5 + 6476*x^6 + ... 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 *) a[ n_] := If[ n<-4, 0, (4^(n + 1) - Binomial[2 n + 2, n + 1]) / 2]; (* Michael Somos, Jan 25 2014 *) PROG (PARI) {a(n) = if( n<-4, 0, n++; (2^(2*n) - binomial(2*n, n)) / 2)}; /* Michael Somos, Jan 25 2014 */ (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. Subdiagonal of A053979. Bisection of A058622 and (possibly) A007008. Cf. A045621, A068551. Cf. A033504, A038806, A055217, A069731. Even bisection of A294175 (without the first two terms). The following relate to compositions of 2n with alternating sum k. - The k > 0 case is counted by A000302. - The k <= 0 case is counted by A000302. - The k != 0 case is counted by A000346 (this sequence). - The k = 0 case is counted by A001700 or A088218. - The k < 0 case is counted by A008549. - The k >= 0 case is counted by A114121. A011782 counts compositions. A086543 counts partitions with nonzero alternating sum (bisection: A182616). A097805 counts compositions by alternating (or reverse-alternating) sum. A103919 counts partitions by sum and alternating sum (reverse: A344612). A345197 counts compositions by length and alternating sum. Cf. A000070, A001791, A007318, A025047, A027306, A032443, A053754, A120452, A163493, A239830, A344611, A345921. Sequence in context: A010036 A127617 A095932 * A289798 A026672 A049652 Adjacent sequences:  A000343 A000344 A000345 * A000347 A000348 A000349 KEYWORD nonn,easy AUTHOR EXTENSIONS Corrected by Christian G. Bower STATUS approved

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Last modified August 5 06:18 EDT 2021. Contains 346457 sequences. (Running on oeis4.)