A066183 Total sum of squares of parts in all partitions of n.

1, 6, 17, 44, 87, 180, 311, 558, 910, 1494, 2302, 3608, 5343, 7986, 11554, 16714, 23549, 33270, 45942, 63506, 86338, 117156, 156899, 209926, 277520, 366260, 479012, 624956, 808935, 1044994, 1340364, 1715572, 2182935, 2770942, 3499379

Offset: 1

Author

Wouter Meeussen, Dec 15 2001

Keywords

• nonn

Comments

Sum of hook lengths of all boxes in the Ferrers diagrams of all partitions of n (see the Guo-Niu Han paper, p. 25, Corollary 6.5). Example: a(3) = 17 because for the partitions (3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1}, {3,2,1}, respectively; the total sum of all hook lengths is 6+5+6 = 17. - Emeric Deutsch, May 15 2008

Partial sums of A206440. - Omar E. Pol, Feb 08 2012

Column k=2 of A213191. - Alois P. Heinz, Sep 20 2013

Row sums of triangles A180681, A206561 and A299768. - Omar E. Pol, Mar 20 2018

Examples

			a(3) = 17 because the squares of all partitions of 3 are {9}, {4,1} and {1,1,1}, summing to 17.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO], May 09 2008.

Crossrefs

Cf. A000041, A001157, A180681, A206440, A206561, A213191, A263004, A265245, A299768.

Programs

Maple

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else g:= b(n, i-1); h:= b(n-i, i);
         [g[1]+h[1], g[2]+h[2] +h[1]*i^2]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2012
# second Maple program:
g := (sum(k^2*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

Mathematica

Table[Apply[Plus, IntegerPartitions[n]^2, {0, 2}], {n, 30}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, g = b[n, i-1]; h = b[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + h[[1]]*i^2}]]; a[n_] :=  b[n, n][[2]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 31 2015, after Alois P. Heinz *)

PARI

a(n)=my(s); forpart(v=n,s+=sum(i=1,#v,v[i]^2));s \\ Charles R Greathouse IV, Aug 31 2015

PARI

a(n)=sum(k=1,n,sigma(k,2)*numbpart(n-k)) \\ Charles R Greathouse IV, Aug 31 2015

Formula

a(n) = Sum_{k=1..n} sigma_2(k)*numbpart(n-k), where sigma_2(k)=sum of squares of divisors of k=A001157(k). - Vladeta Jovovic, Jan 26 2002

a(n) = Sum_{k>=0} k*A265245(n,k). - Emeric Deutsch, Dec 06 2015

G.f.: g(x) = (Sum_{k>=1} k^2*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 3*sqrt(2)*Zeta(3)/Pi^3 * exp(Pi*sqrt(2*n/3)) * sqrt(n). - Vaclav Kotesovec, May 28 2018

Extensions

More terms from Naohiro Nomoto, Feb 07 2002