A004987 - OEIS

%I #90 Nov 13 2024 17:20:00

%S 1,3,18,126,945,7371,58968,480168,3961386,33011550,277297020,

%T 2344420260,19927572210,170150808870,1458435504600,12542545339560,

%U 108179453553705,935434098376155,8107095519260010,70403724246205350,612512400941986545,5337608065351597035,46582761297613937760

%N a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k + 1). 3-central binomial coefficients.

%C Diagonal of rational function R(x,y) = (1 - 9*x*y) / (1 - 2*x - 3*y + 3*y^2 + 9*x^2*y). - _Gheorghe Coserea_, Jul 01 2016

%C This is the k = 3 variant of the k-central binomial coefficients c(n,k) with g.f. (1 - k^2*x)^(-1/k), which yield the usual central binomial coefficients A001405 for k = 2. - _M. F. Hasler_, Nov 12 2024

%H Gheorghe Coserea, <a href="/A004987/b004987.txt">Table of n, a(n) for n = 0..200</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

%H A. Straub, V. H. Moll, and T. Amdeberhan, <a href="http://dx.doi.org/10.4064/aa140-1-2">The p-adic valuation of k-central binomial coefficients</a>, Acta Arith. 140 (2009) 31-41, eq (1.10).

%F G.f.: (1 - 9*x)^(-1/3).

%F a(n) = (3^n/n!)*A007559(n), n >= 1, a(0) := 1.

%F a(n) ~ Gamma(1/3)^-1*n^(-2/3)*3^(2*n)*{1 - 1/9*n^-1 + ...}.

%F Representation as n-th moment of a positive function on (0, 9): a(n) = Integral_{x=0..9} ( x^n*(1/(Pi*sqrt(3)*6*(x/9)^(2/3)*(1-x/9)^(1/3))) ), n >= 0. This function is the solution of the Hausdorff moment problem on (0, 9) with moments equal to a(n). As a consequence this representation is unique. - _Karol A. Penson_, Jan 30 2003

%F D-finite with recurrence: n*a(n) + 3*(2-3*n)*a(n-1)=0. - _R. J. Mathar_, Jun 07 2013

%F 0 = a(n) * (81*a(n+1) - 15*a(n+2)) + a(n+1) * (-3*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Jan 27 2014

%F G.f. A(x)=:y satisfies 0 = y'' * y - 4 * y' * y'. - _Michael Somos_, Jan 27 2014

%F a(n) = (-9)^n*binomial(-1/3, n). - _Peter Luschny_, Mar 23 2014

%F E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([1/3], [1], 9*x). - _Karol A. Penson_, Dec 19 2015

%F Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi + 3*(12 + log(3)))/32 = 1.3980385924595932... - _Ilya Gutkovskiy_, Jul 01 2016

%F Binomial transform of A216316. - _Peter Bala_, Jul 02 2023

%F From _Peter Bala_, Mar 31 2024: (Start)

%F a(n) = (9^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/3, k)* binomial(-1/3, 2*n - k).

%F (9^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

%F Sum_{k = 0..n} a(k)*a(n-k) = A004988(n).

%F Sum_{k = 0..2*n} a(k)*a(2*n-k) = 18^n/(2*n)! * Product_{k = 1..n} (6*k - 1)*(3*k - 2). (End)

%e G.f.: 1 + 3*x + 18*x^2 + 126*x^3 + 945*x^4 + 7371*x^5 + 58968*x^6 + 480168*x^7 + ...

%p a:= n-> (3^n/n!)*mul(3*k+1, k=0..n-1); seq(a(n), n=0..25); # _G. C. Greubel_, Aug 22 2019

%t Table[(-9)^n Binomial[-1/3, n], {n, 0, 25}] (* _Jean-François Alcover_, Sep 28 2016, after _Peter Luschny_ *)

%o (PARI) a(n) = prod(k=0, n-1, 3*k + 1)*3^n/n! \\ _Michel Marcus_, Jun 30 2013

%o (PARI)

%o my(x='x, y='y);

%o R = (1 - 9*x*y) / (1 - 2*x - 3*y + 3*y^2 + 9*x^2*y);

%o diag(n, expr, var) = {

%o   my(a = vector(n));

%o   for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));

%o   for (k = 1, n, a[k] = expr;

%o        for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));

%o   return(a);

%o };

%o diag(20, R, [x,y])  \\ _Gheorghe Coserea_, Jul 01 2016

%o (PARI) Vec((1-9*x+O(x^25))^(-1/3)) \\ yields the same as:

%o apply( {A004987(n)=prod(k=0, n-1, 9*k+3)\n!}, [0..24]) \\ _M. F. Hasler_, Nov 12 2024

%o (Magma) [1] cat [3^n*&*[3*k+1: k in [0..n-1]]/Factorial(n): n in [1..25]]; // _G. C. Greubel_, Aug 22 2019

%o (Sage) [9^n*rising_factorial(1/3, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019

%o (GAP) List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+1)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019

%Y Cf. A004117, A007559, A047657, A004988, A054861, A034689, A053101, A072888, A216316.

%Y Related to diagonal of rational functions: A268545-A268555.

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)

%E More terms from _Ralf Stephan_, Mar 13 2004

%E More terms from _Benoit Cloitre_, Jun 05 2004