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%I #41 Apr 27 2022 17:23:46
%S 0,0,0,0,1,1,2,3,5,8,12,17,25,35,50,68,94,126,170,226,299,391,511,660,
%T 853,1091,1393,1766,2235,2811,3527,4403,5484,6800,8415,10369,12752,
%U 15627,19110,23298,28346,34389,41642,50295,60636,72929,87563,104903,125470
%N Number of 5's in all partitions of n.
%C The sums of five successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C a(n) is also the difference between the sum of 5th largest and the sum of 6th largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%H Alois P. Heinz, <a href="/A024789/b024789.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 a(n) = A181187(n,5) - A181187(n,6). - _Omar E. Pol_, Oct 25 2012
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*Pi*sqrt(2*n)) * (1 - 61*Pi/(24*sqrt(6*n)) + (61/48 + 2521*Pi^2/6912)/n). - _Vaclav Kotesovec_, Nov 05 2016
%F G.f.: x^5/(1 - x^5) * Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Apr 06 2017
%e From _Omar E. Pol_, Oct 25 2012: (Start)
%e For n = 8 we have:
%e --------------------------------------
%e . Number
%e Partitions of 8 of 5's
%e --------------------------------------
%e 8 .............................. 0
%e 4 + 4 .......................... 0
%e 5 + 3 .......................... 1
%e 6 + 2 .......................... 0
%e 3 + 3 + 2 ...................... 0
%e 4 + 2 + 2 ...................... 0
%e 2 + 2 + 2 + 2 .................. 0
%e 7 + 1 .......................... 0
%e 4 + 3 + 1 ...................... 0
%e 5 + 2 + 1 ...................... 1
%e 3 + 2 + 2 + 1 .................. 0
%e 6 + 1 + 1 ...................... 0
%e 3 + 3 + 1 + 1 .................. 0
%e 4 + 2 + 1 + 1 .................. 0
%e 2 + 2 + 2 + 1 + 1 .............. 0
%e 5 + 1 + 1 + 1 .................. 1
%e 3 + 2 + 1 + 1 + 1 .............. 0
%e 4 + 1 + 1 + 1 + 1 .............. 0
%e 2 + 2 + 1 + 1 + 1 + 1 .......... 0
%e 3 + 1 + 1 + 1 + 1 + 1 .......... 0
%e 2 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .. 0
%e ------------------------------------
%e . 7 - 4 = 3
%e The difference between the sum of the fifth column and the sum of the sixth column of the set of partitions of 8 is 7 - 4 = 3 and equals the number of 5's in all partitions of 8, so a(8) = 3.
%e (End)
%p b:= proc(n, i) option remember; local g;
%p if n=0 or i=1 then [1, 0]
%p else g:= `if`(i>n, [0$2], b(n-i, i));
%p b(n, i-1) +g +[0, `if`(i=5, g[1], 0)]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Oct 27 2012
%t Table[ Count[ Flatten[ IntegerPartitions[n]], 5], {n, 1, 50} ]
%t (* second program: *)
%t 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 == 5, 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 (PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(x^5/(1 - x^5) * prod(k=1, 50, 1/(1 - x^k)))) \\ _Indranil Ghosh_, Apr 06 2017
%Y Cf. A066633, A024786, A024787, A024788, A024790, A024791, A024792, A024793, A024794.
%K nonn
%O 1,7
%A _Clark Kimberling_
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