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A024791 Number of 7's in all partitions of n.

%I A024791 #40 Apr 27 2022 17:24:08
%S A024791 0,0,0,0,0,0,1,1,2,3,5,7,11,16,23,32,45,61,84,112,151,199,263,342,446,
%T A024791 574,739,943,1201,1518,1917,2404,3010,3749,4661,5766,7122,8759,10753,
%U A024791 13153,16059,19544,23743,28759,34774,41938,50491,60642,72718,87004,103934,123908
%N A024791 Number of 7's in all partitions of n.
%C A024791 The sums of seven successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C A024791 a(n) is also the difference between the sum of 7th largest and the sum of 8th largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%H A024791 Alois P. Heinz, <a href="/A024791/b024791.txt">Table of n, a(n) for n = 1..1000</a>
%H A024791 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 A024791 a(n) = A181187(n,7) - A181187(n,8). - _Omar E. Pol_, Oct 25 2012
%F A024791 a(n) ~ exp(Pi*sqrt(2*n/3)) / (14*Pi*sqrt(2*n)) * (1 - 85*Pi/(24*sqrt(6*n)) + (85/48 + 4873*Pi^2/6912)/n). - _Vaclav Kotesovec_, Nov 05 2016
%F A024791 G.f.: x^7/(1 - x^7) * Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Apr 06 2017
%p A024791 b:= proc(n, i) option remember; local g;
%p A024791       if n=0 or i=1 then [1, 0]
%p A024791     else g:= `if`(i>n, [0$2], b(n-i, i));
%p A024791          b(n, i-1) +g +[0, `if`(i=7, g[1], 0)]
%p A024791       fi
%p A024791     end:
%p A024791 a:= n-> b(n, n)[2]:
%p A024791 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 27 2012
%t A024791 << DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 7], {n, 1, 52} ]
%t A024791 Table[Count[Flatten[IntegerPartitions[n]],7],{n,55}] (* _Harvey P. Dale_, Feb 26 2015 *)
%t A024791 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 == 7, 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_ *)
%o A024791 (PARI) x='x+O('x^50); concat([0, 0, 0, 0, 0, 0], Vec(x^7/(1 - x^7) * prod(k=1, 50, 1/(1 - x^k)))) \\ _Indranil Ghosh_, Apr 06 2017
%Y A024791 Cf. A066633, A024786, A024787, A024788, A024789, A024790, A024792, A024793, A024794.
%K A024791 nonn
%O A024791 1,9
%A A024791 _Clark Kimberling_