A024793 - OEIS

A024793

Number of 9's in all partitions of n.

%I #35 Apr 27 2022 17:32:48

%S 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 416,533,685,872,1108,1397,1762,2204,2755,3426,4251,5250,6476,7950,

%U 9746,11905,14514,17638,21403,25888,31265,37661,45288,54329,65079,77775

%N Number of 9's in all partitions of n.

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

%C 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 Alois P. Heinz, <a href="/A024793/b024793.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,9) - A181187(n,10). - _Omar E. Pol_, Oct 25 2012

%F From _Peter Bala_, Dec 26 2013: (Start)

%F a(n+9) - a(n) = A000041(n).

%F a(n) + a(n+3) + a(n+6) = A024787(n).

%F 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 Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)

%F 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 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=9, 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]], 9], {n, 1, 55} ]

%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 == 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 Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024794.

%K nonn,easy

%O 1,11

%A _Clark Kimberling_