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A024790 Number of 6's in all partitions of n.

%I A024790 #38 Apr 27 2022 17:24:00
%S A024790 0,0,0,0,0,1,1,2,3,5,7,12,16,24,33,47,63,89,117,159,209,278,360,474,
%T A024790 607,786,1001,1280,1615,2049,2565,3222,4011,4998,6180,7653,9407,11571,
%U A024790 14154,17308,21063,25630,31044,37586,45339,54646,65646,78804,94305,112761,134473
%N A024790 Number of 6's in all partitions of n.
%C A024790 The sums of six successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C A024790 a(n) is also the difference between the sum of 6th largest and the sum of 7th largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%H A024790 Alois P. Heinz, <a href="/A024790/b024790.txt">Table of n, a(n) for n = 1..1000</a>
%H A024790 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 A024790 a(n) = A181187(n,6) - A181187(n,7). - _Omar E. Pol_, Oct 25 2012
%F A024790 From _Peter Bala_, Dec 26 2013: (Start)
%F A024790 a(n+6) - a(n) = A000041(n). a(n) + a(n+3) = A024787(n).
%F A024790 a(n) + a(n+2) + a(n+4) = A024786(n).
%F A024790 O.g.f.: x^6/(1 - x^6) * product {k >= 1} 1/(1 - x^k) = x^6 + x^7 + 2*x^8 + 3*x^9 + ....
%F A024790 Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
%F A024790 a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*Pi*sqrt(2*n)) * (1 - 73*Pi/(24*sqrt(6*n)) + (73/48 + 3601*Pi^2/6912)/n). - _Vaclav Kotesovec_, Nov 05 2016
%p A024790 b:= proc(n, i) option remember; local g;
%p A024790       if n=0 or i=1 then [1, 0]
%p A024790     else g:= `if`(i>n, [0$2], b(n-i, i));
%p A024790          b(n, i-1) +g +[0, `if`(i=6, g[1], 0)]
%p A024790       fi
%p A024790     end:
%p A024790 a:= n-> b(n, n)[2]:
%p A024790 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 27 2012
%t A024790 Table[ Count[ Flatten[ IntegerPartitions[n]], 6], {n, 1, 52} ]
%t A024790 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 == 6, 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 A024790 Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024791, A024792, A024793, A024794.
%K A024790 nonn,easy
%O A024790 1,8
%A A024790 _Clark Kimberling_