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A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).

%I A008780 #47 Dec 29 2024 17:01:56
%S A008780 1,11,48,141,331,672,1232,2094,3357,5137,7568,10803,15015,20398,27168,
%T A008780 35564,45849,58311,73264,91049,112035,136620,165232,198330,236405,
%U A008780 279981,329616,385903,449471,520986,601152,690712,790449,901187,1023792,1159173,1308283
%N A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).
%C A008780 These are the conjectured numbers of d-dimensional partitions for n=6, coming from a formula proposed by MacMahon in the general case that turned out to be wrong. Still, here for n=6, MacMahon's formula coincides for d < 3 with the first three terms of A042984. - _Michel Marcus_, Aug 16 2013
%C A008780 Binomial transform of [1,10,27,29,12,1,0,0,0,...], 6th row of A116672. - _R. J. Mathar_, Jul 18 2017
%D A008780 G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.
%H A008780 Vincenzo Librandi, <a href="/A008780/b008780.txt">Table of n, a(n) for n = 0..1000</a>
%H A008780 S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, <a href="http://arxiv.org/abs/1105.6231">On the asymptotics of higher-dimensional partitions</a>, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
%H A008780 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A008780 G.f.: (1 + 5*x - 3*x^2 - 2*x^3)/(1-x)^6. - _Colin Barker_, Sep 05 2012
%F A008780 From _G. C. Greubel_, Sep 11 2019: (Start)
%F A008780 a(n) = (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120.
%F A008780 E.g.f.: (120 + 1200*x + 1620*x^2 + 580*x^3 + 60*x^4 + x^5)*exp(x)/120. (End)
%p A008780 seq(1+10*n+27*binomial(n,2)+29*binomial(n,3)+12*binomial(n,4)+binomial(n,5), n=0..40);
%t A008780 Table[1+10n+27Binomial[n,2]+29Binomial[n,3]+12Binomial[n,4]+ Binomial[n,5], {n,0,40}] (* _Harvey P. Dale_, Jul 27 2011 *)
%t A008780 CoefficientList[Series[(1+5x-3x^2-2x^3)/(1-x)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 17 2013 *)
%t A008780 LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,48,141,331,672},40] (* _Harvey P. Dale_, Aug 28 2019 *)
%o A008780 (PARI) my(x='x+O('x^40)); Vec((1+5*x-3*x^2-2*x^3)/(1-x)^6) \\ _G. C. Greubel_, Sep 11 2019
%o A008780 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+5*x-3*x^2-2*x^3)/(1-x)^6 )); // _G. C. Greubel_, Sep 11 2019
%o A008780 (Sage) [(120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120 for n in (0..40)] # _G. C. Greubel_, Sep 11 2019
%o A008780 (GAP) List([0..40], n-> (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120); # _G. C. Greubel_, Sep 11 2019
%Y A008780 Cf. A007326, A007327, A042984, A116672.
%K A008780 nonn,nice,easy
%O A008780 0,2
%A A008780 _N. J. A. Sloane_
%E A008780 Description corrected by _Alford Arnold_, Aug 1998
%E A008780 More terms added by _G. C. Greubel_, Sep 11 2019