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A026816 Number of partitions of n in which the greatest part is 10.

%I A026816 #54 Feb 27 2024 10:50:18
%S A026816 0,0,0,0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,22,30,42,55,75,97,128,164,212,
%T A026816 267,340,423,530,653,807,984,1204,1455,1761,2112,2534,3015,3590,4242,
%U A026816 5013,5888,6912,8070,9418,10936,12690,14663,16928,19466
%N A026816 Number of partitions of n in which the greatest part is 10.
%H A026816 Seiichi Manyama, <a href="/A026816/b026816.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H A026816 <a href="/index/Rec#order_55">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).
%F A026816 G.f.: x^10 / (Product_{k=1..10} 1-x^k ). - _Colin Barker_, Feb 22 2013
%F A026816 a(n) = A008284(n,10). - _Robert A. Russell_, May 13 2018
%F A026816 a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} 1. - _Wesley Ivan Hurt_, Jul 13 2019
%t A026816 Table[ Length[ Select[ Partitions[n], First[ # ] == 10 & ]], {n, 1, 60} ]
%t A026816 CoefficientList[Series[x^10/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8) (1 - x^9) (1 - x^10)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%o A026816 (PARI) concat(vector(9),Vec(1/prod(k=1,10,1-x^k)+O(x^90))) \\ _Charles R Greathouse IV_, May 06 2015
%o A026816 (GAP) List([0..70],n->NrPartitions(n,10)); # _Muniru A Asiru_, May 17 2018
%o A026816 (Magma) [#Partitions(k, 10): k in [1..51]]; // _Marius A. Burtea_, Jul 13 2019
%Y A026816 Essentially same as A008639.
%Y A026816 Cf. A026810, A026811, A026812, A026813, A026814, A026815.
%K A026816 nonn,easy
%O A026816 0,13
%A A026816 _Clark Kimberling_
%E A026816 a(0)=0 prepended by _Seiichi Manyama_, Jun 08 2017