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

%I A066183 #71 Jun 01 2018 19:11:23
%S A066183 1,6,17,44,87,180,311,558,910,1494,2302,3608,5343,7986,11554,16714,
%T A066183 23549,33270,45942,63506,86338,117156,156899,209926,277520,366260,
%U A066183 479012,624956,808935,1044994,1340364,1715572,2182935,2770942,3499379
%N A066183 Total sum of squares of parts in all partitions of n.
%C A066183 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 A066183 Partial sums of A206440. - _Omar E. Pol_, Feb 08 2012
%C A066183 Column k=2 of A213191. - _Alois P. Heinz_, Sep 20 2013
%C A066183 Row sums of triangles A180681, A206561 and A299768. - _Omar E. Pol_, Mar 20 2018
%H A066183 Alois P. Heinz, <a href="/A066183/b066183.txt">Table of n, a(n) for n = 1..10000</a>
%H A066183 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 A066183 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 A066183 a(n) = Sum_{k>=0} k*A265245(n,k). - _Emeric Deutsch_, Dec 06 2015
%F A066183 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 A066183 a(n) ~ 3*sqrt(2)*Zeta(3)/Pi^3 * exp(Pi*sqrt(2*n/3)) * sqrt(n). - _Vaclav Kotesovec_, May 28 2018
%e A066183 a(3) = 17 because the squares of all partitions of 3 are {9}, {4,1} and {1,1,1}, summing to 17.
%p A066183 b:= proc(n, i) option remember; local g, h;
%p A066183       if n=0 then [1, 0]
%p A066183     elif i<1 then [0, 0]
%p A066183     elif i>n then b(n, i-1)
%p A066183     else g:= b(n, i-1); h:= b(n-i, i);
%p A066183          [g[1]+h[1], g[2]+h[2] +h[1]*i^2]
%p A066183       fi
%p A066183     end:
%p A066183 a:= n-> b(n, n)[2]:
%p A066183 seq(a(n), n=1..40);  # _Alois P. Heinz_, Feb 23 2012
%p A066183 # second Maple program:
%p A066183 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 A066183 Table[Apply[Plus, IntegerPartitions[n]^2, {0, 2}], {n, 30}]
%t A066183 (* Second program: *)
%t A066183 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 A066183 (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 A066183 (PARI) a(n)=sum(k=1,n,sigma(k,2)*numbpart(n-k)) \\ _Charles R Greathouse IV_, Aug 31 2015
%Y A066183 Cf. A000041, A001157, A180681, A206440, A206561, A213191, A263004, A265245, A299768.
%K A066183 nonn
%O A066183 1,2
%A A066183 _Wouter Meeussen_, Dec 15 2001
%E A066183 More terms from _Naohiro Nomoto_, Feb 07 2002