A327731 - OEIS

%I #6 Sep 23 2019 15:32:45

%S 1,1,1,3,3,5,8,10,14,21,28,36,51,65,86,117,148,190,251,316,402,519,

%T 647,814,1032,1282,1593,1994,2457,3029,3754,4591,5617,6895,8381,10193,

%U 12411,14999,18125,21919,26359,31672,38074,45556,54468,65134,77576,92322

%N Expansion of Product_{i>=1, j>=1}  (1 + x^(i*(2*j - 1))).

%C Weigh transform of A001227.

%F G.f.: Product_{k>=1} (1 + x^k)^A001227(k).

%t nmax = 47; CoefficientList[Series[Product[(1 + x^k)^DivisorSum[k, Mod[#, 2] &], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSum[d, Mod[#, 2] &], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

%o (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^numdiv(k>>valuation(k, 2))))} \\ _Andrew Howroyd_, Sep 23 2019

%Y Cf. A001227, A107742, A301800, A320650.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Sep 23 2019