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A024792
Number of 8's in all partitions of n.
12
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 44, 59, 82, 108, 146, 191, 254, 328, 429, 549, 709, 900, 1148, 1446, 1829, 2286, 2865, 3559, 4427, 5465, 6752, 8288, 10178, 12429, 15175, 18442, 22404, 27102, 32767, 39473, 47516, 57012, 68349, 81703, 97579, 116236
OFFSET
1,10
COMMENTS
The sums of eight successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 8th largest and the sum of 9th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
LINKS
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
FORMULA
a(n) = A181187(n,8) - A181187(n,9). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+8) - a(n) = A000041(n). a(n) + a(n+4) = A024788(n).
a(n) + a(n+2) + a(n+4) + a(n+6) = A024786(n).
O.g.f.: x^8/(1 - x^8) * product {k >= 1} 1/(1 - x^k) = x^8 + x^9 + 2*x^10 + 3*x^11 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*Pi*sqrt(2*n)) * (1 - 97*Pi/(24*sqrt(6*n)) + (97/48 + 6337*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
MAPLE
b:= proc(n, i) option remember; local g;
if n=0 or i=1 then [1, 0]
else g:= `if`(i>n, [0$2], b(n-i, i));
b(n, i-1) +g +[0, `if`(i=8, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100); # Alois P. Heinz, Oct 27 2012
MATHEMATICA
Table[ Count[ Flatten[ IntegerPartitions[n]], 8], {n, 1, 53} ]
(* second program: *)
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 == 8, 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 *)
KEYWORD
nonn,easy
STATUS
approved