A024789 Number of 5's in all partitions of n.

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

The sums of five successive terms give A000070. - Omar E. Pol, Jul 12 2012

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

Examples

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 8 we have:
--------------------------------------
.                             Number
Partitions of 8               of 5's
--------------------------------------
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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Crossrefs

Cf. A066633, A024786, A024787, A024788, A024790, A024791, A024792, A024793, A024794.

Programs

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=5, 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]], 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 *)

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

Formula

a(n) = A181187(n,5) - A181187(n,6). - Omar E. Pol, Oct 25 2012

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