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 A066183 Total sum of squares of parts in all partitions of n. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 a(n) = Sum(k*A265245(n,k), k>=0). - Emeric Deutsch, Dec 06 2015 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. 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 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 EXAMPLE a(3) = 17 because the squares of all partitions of 3 are {9}, {4,1} and {1,1,1}, summing to 17. 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 *) PROG (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 CROSSREFS Cf. A000041, A001157, A265245. Sequence in context: A099858 A232567 A062020 * A262297 A048746 A026382 Adjacent sequences:  A066180 A066181 A066182 * A066184 A066185 A066186 KEYWORD nonn AUTHOR Wouter Meeussen, Dec 15 2001 EXTENSIONS More terms from Naohiro Nomoto, Feb 07 2002 STATUS approved

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