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A024789
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Number of 5's in all partitions of n.
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13
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0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 68, 94, 126, 170, 226, 299, 391, 511, 660, 853, 1091, 1393, 1766, 2235, 2811, 3527, 4403, 5484, 6800, 8415, 10369, 12752, 15627, 19110, 23298, 28346, 34389, 41642, 50295, 60636, 72929, 87563, 104903, 125470
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OFFSET
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1,7
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COMMENTS
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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
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LINKS
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FORMULA
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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
G.f.: x^5/(1 - x^5) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 06 2017
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EXAMPLE
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For n = 8 we have:
--------------------------------------
. Number
Partitions of 8 of 5's
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8 .............................. 0
4 + 4 .......................... 0
5 + 3 .......................... 1
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 0
2 + 2 + 2 + 2 .................. 0
7 + 1 .......................... 0
4 + 3 + 1 ...................... 0
5 + 2 + 1 ...................... 1
3 + 2 + 2 + 1 .................. 0
6 + 1 + 1 ...................... 0
3 + 3 + 1 + 1 .................. 0
4 + 2 + 1 + 1 .................. 0
2 + 2 + 2 + 1 + 1 .............. 0
5 + 1 + 1 + 1 .................. 1
3 + 2 + 1 + 1 + 1 .............. 0
4 + 1 + 1 + 1 + 1 .............. 0
2 + 2 + 1 + 1 + 1 + 1 .......... 0
3 + 1 + 1 + 1 + 1 + 1 .......... 0
2 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 .. 0
------------------------------------
. 7 - 4 = 3
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.
(End)
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MAPLE
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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=5, g[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
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MATHEMATICA
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Table[ Count[ Flatten[ IntegerPartitions[n]], 5], {n, 1, 50} ]
(* 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 == 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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