A024794 Number of 10's in all partitions of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 79, 104, 140, 183, 242, 312, 407, 520, 670, 849, 1081, 1359, 1715, 2141, 2678, 3322, 4125, 5085, 6274, 7691, 9430, 11502, 14025, 17024, 20655, 24959, 30140, 36270, 43612, 52274, 62604, 74763

Offset

1,12

Comments

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

a(n) is also the difference between the sum of 10th largest and the sum of 11th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

In general, if m>0 and a(n+m)-a(n) = A000041(n), then a(n) ~ exp(sqrt(2*n/3)*Pi) / (2*Pi*m*sqrt(2*n)) * (1 - Pi*(1/24 + m/2)/sqrt(6*n) + (1/48 + Pi^2/6912 + m/4 + m*Pi^2/288 + m^2*Pi^2/72)/n). - Vaclav Kotesovec, Nov 05 2016

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.

Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.

Formula

a(n) = A181187(n,10) - A181187(n,11). - Omar E. Pol, Oct 25 2012

From Peter Bala, Dec 26 2013: (Start)

a(n+10) - a(n) = A000041(n). a(n) + a(n+5) = A024789(n).

a(n) + a(n+2) + a(n+4) + a(n+6) + a(n+8) = A024786(n).

O.g.f.: x^10/(1 - x^10) * product {k >= 1} 1/(1 - x^k) = x^10 + x^11 + 2*x^12 + 3*x^13 + ....

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

a(n) ~ exp(Pi*sqrt(2*n/3)) / (20*Pi*sqrt(2*n)) * (1 - 121*Pi/(24*sqrt(6*n)) + (121/48 + 9841*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=10, 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]], 10], {n, 1, 55} ]
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 == 10, 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 *)

Crossrefs

Cf. A066633, A000070(n-1), A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A000041.

Sequence in context: A218026 A241728 A326589 * A326292 A195308 A218025

Adjacent sequences: A024791 A024792 A024793 * A024795 A024796 A024797

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved