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A024791 Number of 7's in all partitions of n. 12

%I #40 Apr 27 2022 17:24:08

%S 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 574,739,943,1201,1518,1917,2404,3010,3749,4661,5766,7122,8759,10753,

%U 13153,16059,19544,23743,28759,34774,41938,50491,60642,72718,87004,103934,123908

%N Number of 7's in all partitions of n.

%C The sums of seven successive terms give A000070. - _Omar E. Pol_, Jul 12 2012

%C 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 Alois P. Heinz, <a href="/A024791/b024791.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,7) - A181187(n,8). - _Omar E. Pol_, Oct 25 2012

%F 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 G.f.: x^7/(1 - x^7) * Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Apr 06 2017

%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=7, 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 << DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 7], {n, 1, 52} ]

%t Table[Count[Flatten[IntegerPartitions[n]],7],{n,55}] (* _Harvey P. Dale_, Feb 26 2015 *)

%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 == 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 (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 Cf. A066633, A024786, A024787, A024788, A024789, A024790, A024792, A024793, A024794.

%K nonn

%O 1,9

%A _Clark Kimberling_

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)