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%I #47 Sep 08 2022 08:45:46
%S 5,35,105,231,429,715,1105,1615,2261,3059,4025,5175,6525,8091,9889,
%T 11935,14245,16835,19721,22919,26445,30315,34545,39151,44149,49555,
%U 55385,61655,68381,75579,83265,91455,100165,109411,119209,129575,140525,152075,164241
%N a(n) = (2*n+1)*(2*n+3)*(2*n+5)/3.
%H Vincenzo Librandi, <a href="/A162540/b162540.txt">Table of n, a(n) for n = 0..10000</a>
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H M. R. Sepanski, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p32">On Divisibility of Convolutions of Central Binomial Coefficients</a>, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A061550(n)/3 = A077415(2*n+3).
%F From _R. J. Mathar_, Jul 16 2009: (Start)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
%F G.f.: (5 + 15*x - 5*x^2 + x^3)/(x-1)^4. (End)
%F a(n) = 5*Pochhammer(7/2,n)/Pochhammer(1/2,n). Hence e.g.f. is 5* 1F1(7/2;1/2;x), with 1F1 being the confluent hypergemetric function (also known as Kummer's). - _Stanislav Sykora_, May 26 2016
%F E.g.f.: (8*x^3 + 60*x^2 + 90*x + 15)*exp(x)/3. - _Robert Israel_, May 27 2016
%F From _Amiram Eldar_, Jan 09 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 1/4.
%F Sum_{n>=0} (-1)^n/a(n) = 3*Pi/8 - 1 = A093828 - 1. (End)
%p A162540:=n->(2*n+1)*(2*n+3)*(2*n+5)/3: seq(A162540(n), n=0..80); # _Wesley Ivan Hurt_, May 28 2016
%t Table[((2n+1)(2n+3)(2n+5))/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{5,35,105,231},40] (* _Harvey P. Dale_, Nov 06 2011 *)
%o (Magma) [(2*n+1)*(2*n+3)*(2*n+5)/3: n in [0..40]]; // _Vincenzo Librandi_, Nov 16 2011
%o (PARI) Vec((5+15*x-5*x^2+x^3)/(x-1)^4 + O(x^100)) \\ _Altug Alkan_, Oct 26 2015
%Y Cf. A061550, A077415, A093828.
%K nonn,easy
%O 0,1
%A Jacob Landon (jacoblandon(AT)aol.com), Jul 05 2009
%E Offset corrected, definition clarified by _R. J. Mathar_, Jul 16 2009
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