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A024793 Number of 9's in all partitions of n.

%I A024793 #35 Apr 27 2022 17:32:48
%S A024793 0,0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,22,31,43,58,80,106,142,187,246,319,
%T A024793 416,533,685,872,1108,1397,1762,2204,2755,3426,4251,5250,6476,7950,
%U A024793 9746,11905,14514,17638,21403,25888,31265,37661,45288,54329,65079,77775
%N A024793 Number of 9's in all partitions of n.
%C A024793 The sums of nine successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C A024793 a(n) is also the difference between the sum of 9th largest and the sum of 10th largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%H A024793 Alois P. Heinz, <a href="/A024793/b024793.txt">Table of n, a(n) for n = 1..1000</a>
%H A024793 David Benson, Radha Kessar, and Markus Linckelmann, <a href="https://arxiv.org/abs/2204.09970">Hochschild cohomology of symmetric groups in low degrees</a>, arXiv:2204.09970 [math.GR], 2022.
%F A024793 a(n) = A181187(n,9) - A181187(n,10). - _Omar E. Pol_, Oct 25 2012
%F A024793 From _Peter Bala_, Dec 26 2013: (Start)
%F A024793 a(n+9) - a(n) = A000041(n).
%F A024793 a(n) + a(n+3) + a(n+6) = A024787(n).
%F A024793 O.g.f.: x^9/(1 - x^9) * product {k >= 1} 1/(1 - x^k) = x^9 + x^10 + 2*x^11 + 3*x^12 + ....
%F A024793 Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
%F A024793 a(n) ~ exp(Pi*sqrt(2*n/3)) / (18*Pi*sqrt(2*n)) * (1 - 109*Pi/(24*sqrt(6*n)) + (109/48 + 7993*Pi^2/6912)/n). - _Vaclav Kotesovec_, Nov 05 2016
%p A024793 b:= proc(n, i) option remember; local g;
%p A024793       if n=0 or i=1 then [1, 0]
%p A024793     else g:= `if`(i>n, [0$2], b(n-i, i));
%p A024793          b(n, i-1) +g +[0, `if`(i=9, g[1], 0)]
%p A024793       fi
%p A024793     end:
%p A024793 a:= n-> b(n, n)[2]:
%p A024793 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 27 2012
%t A024793 Table[ Count[ Flatten[ IntegerPartitions[n]], 9], {n, 1, 55} ]
%t A024793 (* second program: *)
%t A024793 b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 9, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 09 2015, after _Alois P. Heinz_ *)
%Y A024793 Cf.  A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024794.
%K A024793 nonn,easy
%O A024793 1,11
%A A024793 _Clark Kimberling_