A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

0, 1, 3, 7, 16, 28, 59, 91, 170, 269, 450, 655, 1162, 1602, 2527, 3793, 5805, 8034, 12660, 17131, 26484, 37384, 53738, 73504, 114683, 153613, 221225, 313339, 453769, 609179, 927968, 1223909, 1804710, 2522264, 3539835, 4855420, 7439870, 9765555, 14009545

Offset: 0

Author

Alois P. Heinz, Feb 27 2013

Keywords

• nonn

Examples

			a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.

Programs

Maple

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

Mathematica

b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

Formula

From Vaclav Kotesovec, May 24 2018: (Start)

a(n) ~ c * 3^(n/3), where

c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0

c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1

c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2

(End)