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Total sum of squares of parts in all partitions of n.
15

%I #71 Jun 01 2018 19:11:23

%S 1,6,17,44,87,180,311,558,910,1494,2302,3608,5343,7986,11554,16714,

%T 23549,33270,45942,63506,86338,117156,156899,209926,277520,366260,

%U 479012,624956,808935,1044994,1340364,1715572,2182935,2770942,3499379

%N Total sum of squares of parts in all partitions of n.

%C 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

%C Partial sums of A206440. - _Omar E. Pol_, Feb 08 2012

%C Column k=2 of A213191. - _Alois P. Heinz_, Sep 20 2013

%C Row sums of triangles A180681, A206561 and A299768. - _Omar E. Pol_, Mar 20 2018

%H Alois P. Heinz, <a href="/A066183/b066183.txt">Table of n, a(n) for n = 1..10000</a>

%H Guo-Niu Han, <a href="https://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths</a>, arXiv:0804.1849v3 [math.CO], May 09 2008.

%F 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

%F a(n) = Sum_{k>=0} k*A265245(n,k). - _Emeric Deutsch_, Dec 06 2015

%F 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

%F a(n) ~ 3*sqrt(2)*Zeta(3)/Pi^3 * exp(Pi*sqrt(2*n/3)) * sqrt(n). - _Vaclav Kotesovec_, May 28 2018

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

%p b:= proc(n, i) option remember; local g, h;

%p if n=0 then [1, 0]

%p elif i<1 then [0, 0]

%p elif i>n then b(n, i-1)

%p else g:= b(n, i-1); h:= b(n-i, i);

%p [g[1]+h[1], g[2]+h[2] +h[1]*i^2]

%p fi

%p end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Feb 23 2012

%p # second Maple program:

%p 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

%t Table[Apply[Plus, IntegerPartitions[n]^2, {0, 2}], {n, 30}]

%t (* Second program: *)

%t 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_ *)

%o (PARI) a(n)=my(s); forpart(v=n,s+=sum(i=1,#v,v[i]^2));s \\ _Charles R Greathouse IV_, Aug 31 2015

%o (PARI) a(n)=sum(k=1,n,sigma(k,2)*numbpart(n-k)) \\ _Charles R Greathouse IV_, Aug 31 2015

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

%K nonn

%O 1,2

%A _Wouter Meeussen_, Dec 15 2001

%E More terms from _Naohiro Nomoto_, Feb 07 2002