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 A025174 a(n) = binomial(3n-1, n-1). 30
 0, 1, 5, 28, 165, 1001, 6188, 38760, 245157, 1562275, 10015005, 64512240, 417225900, 2707475148, 17620076360, 114955808528, 751616304549, 4923689695575, 32308782859535, 212327989773900, 1397281501935165 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of standard tableaux of shape (2n-1,n). Example: a(2)=5 because in the top row we can have 123, 124, 125, 134, or 135. - Emeric Deutsch, May 23 2004 Number of peaks in all generalized {(1,2),(1,-1)}-Dyck paths of length 3n. REFERENCES B. C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag, see Entry 14, Corollary 1, p. 71. LINKS Robert Israel, Table of n, a(n) for n = 0..1190 D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6. W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013. Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2. FORMULA G.f.: z*g^2/(1-3*z*g^2), where g=g(z) is given by g=1+z*g^3, g(0)=1, that is, (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003 a(n) = sum{k=0..n, ((3k+1)/(2n+k+1))C(3n, 2n+k)*A001045(k)}. - Paul Barry, Oct 07 2005 Hankel transform of a(n+1) is A005156(n+1). - Paul Barry, Apr 14 2008 G.f.: x*B'(x)/B(x) where B(x) is the g.f. of A001764. - Vladimir Kruchinin Feb 03 2013 2*n*(2*n-1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 05 2013 Logarithmic derivative of A001764; g.f. of A001764 satisfies G(x) = 1 + x*G(x)^3. - Paul D. Hanna, Jul 14 2013 G.f.: (2*cos((1/3)*arcsin((3/2)*sqrt(3*x)))-sqrt(4-27*x))/(3*sqrt(4-27*x)). - Emanuele Munarini, Oct 14 2014 a(n) = sum(k=1..n, binomial(n-1,n-k)*binomial(2*n,n-k)). - Vladimir Kruchinin, Nov 12 2014 a(n) = [x^n] C(x)^n for n >= 1, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function for A000108 (Ramanujan). - Peter Bala, Jun 24 2015 From Peter Bala, Nov 04 2015: (Start) Without the initial term 0, the o.g.f. equals f(x)*g(x)^2, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. g(x)^2 is the o.g..f for A006013. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End) EXAMPLE L.g.f.: L(x) = x + 5*x^2/2 + 28*x^3/3 + 165*x^4/4 + 1001*x^5/5 + 6188*x^6/6 + ... where G(x) = exp(L(x)) satisfies G(x) = 1 + x*G(x)^3, and begins: exp(L(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ... + A001764(n)*x^n + ... MAPLE with(combinat):seq(numbcomp(3*i, i), i=0..20); # Zerinvary Lajos, Jun 16 2007 MATHEMATICA Table[ GegenbauerC[ n, n, 1 ]/2, {n, 0, 24} ] PROG (MAGMA) [Binomial(3*n-1, n-1): n in [0..30]]; // Vincenzo Librandi, Nov 12 2014 (PARI) vector(30, n, n--; binomial(3*n-1, n-1)) \\ Altug Alkan, Nov 04 2015 CROSSREFS Cf. A001764 (binomial(3n,n)/(2n+1)), A117671 (C(3n+1,n+1)), A004319, A005809, A006013, A013698, A045721, A117671, A165817, A236194. Sequence in context: A243669 A254538 A090040 * A083316 A027284 A069731 Adjacent sequences:  A025171 A025172 A025173 * A025175 A025176 A025177 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 17 15:32 EDT 2019. Contains 328116 sequences. (Running on oeis4.)