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 A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0). 1
 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,74 COMMENTS Number of entries in row n = 1 + n^2. Sum of entries in row n = A000041(n). Sum(k*T(n,k), k>=0) = A066183(n). LINKS Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008. FORMULA G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k). EXAMPLE Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively. Triangle starts: 1; 0,1; 0,0,1,0,1; 0,0,0,1,0,1,0,0,0,1; 0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1. MAPLE g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form MATHEMATICA m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* Jean-François Alcover, Feb 19 2019 *) CROSSREFS Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253. Sequence in context: A091393 A284557 A182032 * A110270 A284825 A318875 Adjacent sequences:  A265242 A265243 A265244 * A265246 A265247 A265248 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 06 2015 STATUS approved

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Last modified January 25 19:00 EST 2022. Contains 350572 sequences. (Running on oeis4.)