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%I #84 Feb 27 2024 10:42:40
%S 0,0,0,0,0,1,1,2,3,5,7,10,13,18,23,30,37,47,57,70,84,101,119,141,164,
%T 192,221,255,291,333,377,427,480,540,603,674,748,831,918,1014,1115,
%U 1226,1342,1469,1602,1747,1898,2062,2233,2418,2611,2818,3034,3266,3507,3765
%N Number of partitions of n in which the greatest part is 5.
%C Essentially same as A001401: five zeros followed by A001401.
%C Also number of partitions of n into exactly 5 parts.
%D D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.
%H Washington Bomfim, <a href="/A026811/b026811.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).
%F a(n) = round( ((n^4+10*(n^3+n^2)-75*n -45*n*(-1)^n)) / 2880 ). - _Washington Bomfim_, Jul 03 2012
%F G.f.: x^5/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). - _Joerg Arndt_, Jul 04 2012
%F a(n) = A008284(n,5). - _Robert A. Russell_, May 13 2018
%F From _Gregory L. Simay_, Jul 28 2019: (Start)
%F a(2n) = a(2n-1) + a(n+1) + a(n) - a(n-3) - a(n-4);
%F a(2n+1) = a(2n) + a(n+3) - a(n-5). (End)
%F From _R. J. Mathar_, Jun 23 2021: (Start)
%F a(n) - a(n-5) = A001400(n-5).
%F a(n) - a(n-4) = A008669(n-5).
%F a(n) - a(n-3) = A029007(n-5).
%F a(n) - a(n-2) = A029032(n-5).
%F a(n) = +a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15). (End)
%t Table[Count[IntegerPartitions[n], {5, ___}], {n, 0, 55}] (* corrected by _Harvey P. Dale_, Oct 24 2011 *)
%t Table[Length[IntegerPartitions[n, {5}]], {n, 0, 55}] (* _Eric Rowland_, Mar 02 2017 *)
%t CoefficientList[Series[x^5/Product[1 - x^k, {k, 1, 5}], {x, 0, 65}], x] (* _Robert A. Russell_, May 13 2018 *)
%t Drop[LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1}, Append[Table[0,{14}],1],110],9] (* _Robert A. Russell_, May 17 2018 *)
%o (PARI)
%o a(n)=round((n^4+10*(n^3+n^2)-75*n-45*(-1)^n*n)/2880);
%o for(n=0,10000,print(n," ",a(n))); /* b-file format */
%o /* _Washington Bomfim_, Jul 03 2012 */
%o (PARI) x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ _Altug Alkan_, May 17 2018
%o (GAP) List([0..70],n->NrPartitions(n,5)); # _Muniru A Asiru_, May 17 2018
%Y Cf. A026810, A026812, A026813, A026814, A026815, A026816, A002622 (partial sums), A008667 (first differences).
%K nonn,easy
%O 0,8
%A _Clark Kimberling_
%E More terms from _Robert G. Wilson v_, Jan 11 2002
%E a(0)=0 inserted by _Joerg Arndt_, Jul 04 2012
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