# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a162540 Showing 1-1 of 1 %I A162540 #47 Sep 08 2022 08:45:46 %S A162540 5,35,105,231,429,715,1105,1615,2261,3059,4025,5175,6525,8091,9889, %T A162540 11935,14245,16835,19721,22919,26445,30315,34545,39151,44149,49555, %U A162540 55385,61655,68381,75579,83265,91455,100165,109411,119209,129575,140525,152075,164241 %N A162540 a(n) = (2*n+1)*(2*n+3)*(2*n+5)/3. %H A162540 Vincenzo Librandi, Table of n, a(n) for n = 0..10000 %H A162540 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4. %H A162540 M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32. %H A162540 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). %F A162540 a(n) = A061550(n)/3 = A077415(2*n+3). %F A162540 From _R. J. Mathar_, Jul 16 2009: (Start) %F A162540 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. %F A162540 G.f.: (5 + 15*x - 5*x^2 + x^3)/(x-1)^4. (End) %F A162540 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 A162540 E.g.f.: (8*x^3 + 60*x^2 + 90*x + 15)*exp(x)/3. - _Robert Israel_, May 27 2016 %F A162540 From _Amiram Eldar_, Jan 09 2021: (Start) %F A162540 Sum_{n>=0} 1/a(n) = 1/4. %F A162540 Sum_{n>=0} (-1)^n/a(n) = 3*Pi/8 - 1 = A093828 - 1. (End) %p A162540 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 A162540 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 A162540 (Magma) [(2*n+1)*(2*n+3)*(2*n+5)/3: n in [0..40]]; // _Vincenzo Librandi_, Nov 16 2011 %o A162540 (PARI) Vec((5+15*x-5*x^2+x^3)/(x-1)^4 + O(x^100)) \\ _Altug Alkan_, Oct 26 2015 %Y A162540 Cf. A061550, A077415, A093828. %K A162540 nonn,easy %O A162540 0,1 %A A162540 Jacob Landon (jacoblandon(AT)aol.com), Jul 05 2009 %E A162540 Offset corrected, definition clarified by _R. J. Mathar_, Jul 16 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE