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A008779 Number of n-dimensional partitions of 5.

%I A008779 #48 Jan 18 2025 16:36:05
%S A008779 1,7,24,59,120,216,357,554,819,1165,1606,2157,2834,3654,4635,5796,
%T A008779 7157,8739,10564,12655,15036,17732,20769,24174,27975,32201,36882,
%U A008779 42049,47734,53970,60791,68232,76329,85119,94640,104931,116032,127984,140829,154610,169371
%N A008779 Number of n-dimensional partitions of 5.
%C A008779 a(n) = number of (n+8)-bit binary sequences with exactly 8 1's none of which is isolated. - _David Callan_, Jul 15 2004
%C A008779 For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016
%C A008779 Binomial transform of [1,6,11,7,1,0,0,0,...], the 5th row of A116672. - _R. J. Mathar_, Jul 18 2017
%D A008779 G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.
%H A008779 G. C. Greubel, <a href="/A008779/b008779.txt">Table of n, a(n) for n = 0..1000</a>
%H A008779 P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A008779 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%H A008779 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A008779 G.f.: (1 +2*x -x^2 -x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F A008779 a(n) = (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24. - _M. F. Hasler_, Sep 15 2009
%F A008779 a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). - _Vincenzo Librandi_, May 21 2015
%F A008779 E.g.f.: (24 + 144*x + 132*x^2 + 28*x^3 + x^4)*exp(x)/24. - _G. C. Greubel_, Sep 11 2019
%p A008779 seq(1+6*n+11*binomial(n,2)+7*binomial(n,3)+binomial(n,4), n=0..45);
%t A008779 CoefficientList[Series[(1+2*x-x^2-x^3)/(1-x)^5, {x,0,45}], x] (* _Vincenzo Librandi_, May 21 2015 *)
%t A008779 LinearRecurrence[{5,-10,10,-5,1}, {1,7,24,59,120}, 46] (* _G. C. Greubel_, Sep 11 2019 *)
%o A008779 (Magma) [(n+1)*(n^3+21*n^2+38*n+24)/24: n in [0..45]]; // _Vincenzo Librandi_, May 21 2015
%o A008779 (Magma) I:=[1,7,24,59,120]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // _Vincenzo Librandi_, May 21 2015
%o A008779 (PARI) Vec((-1+x^3+x^2-2*x)/(x-1)^5 + O(x^45)) \\ _Altug Alkan_, Jan 07 2016
%o A008779 (Sage) [(n+1)*(n^3 + 21*n^2 + 38*n + 24)/24 for n in (0..45)] # _G. C. Greubel_, Sep 11 2019
%o A008779 (GAP) List([0..45], n-> (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24); # _G. C. Greubel_, Sep 11 2019
%Y A008779 Cf. A116672, A289656.
%K A008779 nonn,easy
%O A008779 0,2
%A A008779 _N. J. A. Sloane_
%E A008779 More terms from _Vincenzo Librandi_, May 21 2015