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A051836 a(n) = n*(n+1)*(n+2)*(n+3)*(3*n+2)/120.

%I A051836 #80 Feb 16 2025 08:32:41
%S A051836 0,1,8,33,98,238,504,966,1716,2871,4576,7007,10374,14924,20944,28764,
%T A051836 38760,51357,67032,86317,109802,138138,172040,212290,259740,315315,
%U A051836 380016,454923,541198,640088,752928,881144,1026256,1189881,1373736,1579641,1809522,2065414
%N A051836 a(n) = n*(n+1)*(n+2)*(n+3)*(3*n+2)/120.
%C A051836 5-dimensional version of pentagonal-based pyramidal numbers. - Ben Creech (mathroxmysox(AT)yahoo.com)
%C A051836 If Y is a 3-subset of an n-set X then, for n>=7, a(n-6) is the number of 7-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Nov 23 2007
%C A051836 Antidiagonal sums of the convolution array A213548. - _Clark Kimberling_, Jun 17 2012
%C A051836 After 0, convolution of nonzero triangular numbers (A000217) and nonzero pentagonal numbers (A000326). - _Bruno Berselli_, Jun 27 2013
%C A051836 a(n) is also the number of odd chordless cycles in the graph complement of the (n+1)-Andrásfai graph. - _Eric W. Weisstein_, Apr 14 2017
%D A051836 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D A051836 Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
%H A051836 Vincenzo Librandi, <a href="/A051836/b051836.txt">Table of n, a(n) for n = 0..1000</a>
%H A051836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndrasfaiGraph.html">Andrásfai Graph</a>.
%H A051836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>.
%H A051836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphComplement.html">Graph Complement</a>.
%H A051836 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%H A051836 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A051836 a(n) = C(n+4, n)*(3n+5)/5.
%F A051836 G.f.: x*(1+2*x)/(1-x)^6. (adapted by _Vincenzo Librandi_, Jul 04 2017)
%F A051836 From _Amiram Eldar_, Feb 15 2022: (Start)
%F A051836 Sum_{n>=1} 1/a(n) = 135*sqrt(3)*Pi/14 - 1215*log(3)/14 + 925/21.
%F A051836 Sum_{n>=1} (-1)^(n+1)/a(n) = 135*sqrt(3)*Pi/7 - 880*log(2)/7 - 355/21. (End)
%F A051836 E.g.f.: (1/5!)*x*(120 + 360*x + 240*x^2 + 50*x^3 + 3*x^4)*exp(x). - _G. C. Greubel_, Dec 27 2023
%e A051836 By the fourth comment: A000217(1..6) and A000326(1..6) give the term a(6) = 1*21+5*15+12*10+22*6+35*3+51*1 = 504. - _Bruno Berselli_, Jun 27 2013
%p A051836 with (combinat):a[0]:=0:for n from 1 to 50 do a[n]:=stirling2(n+2,n)+a[n-1] od: seq(a[n], n=0..34); # _Zerinvary Lajos_, Mar 17 2008
%t A051836 Table[n(n + 1)(n + 2)(n + 3)(3n + 2)/120, {n, 0, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 08 2011 *)
%t A051836 CoefficientList[Series[x (1 + 2 x) / (1 - x)^6, {x, 0, 33}], x] (* _Vincenzo Librandi_, Jul 04 2017 *)
%t A051836 LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,8,33,98,238},40] (* _Harvey P. Dale_, Jun 01 2018 *)
%o A051836 (PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(3*n+2)/120 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A051836 (Magma) [0] cat [Binomial(n+4, n)*(3*n+5)/5: n in [0..40]]; // _Vincenzo Librandi_, Jul 04 2017
%o A051836 (SageMath) [((3*n+2)/(n+4))*binomial(n+4,5) for n in range(41)] # _G. C. Greubel_, Dec 27 2023
%Y A051836 Partial sums of A001296.
%Y A051836 Cf. A093560 ((3, 1) Pascal, column m=5).
%Y A051836 Cf. A000217, A000326, A213548.
%K A051836 nonn,easy
%O A051836 0,3
%A A051836 _Barry E. Williams_, Dec 12 1999
%E A051836 Simpler definition from Ben Creech (mathroxmysox(AT)yahoo.com), Nov 13 2005