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A318844 Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005). 1

%I #7 Apr 03 2019 02:55:54

%S 1,0,1,1,2,2,5,4,8,10,15,17,29,31,48,60,81,99,143,167,231,287,374,460,

%T 615,740,964,1194,1512,1856,2379,2877,3635,4460,5540,6759,8433,10192,

%U 12608,15335,18774,22726,27868,33525,40863,49292,59652,71694,86780,103818,125118,149778,179608

%N Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

%C Convolution of A081362 and A107742.

%C Weigh transform of A032741.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: Product_{k>=1} (1 + x^k)^A032741(k).

%F G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/(k*(1 - x^(2*k)))), where sigma_1(k) = sum of divisors of k (A000203).

%p with(numtheory): a:=series(mul((1+x^k)^(tau(k)-1),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # _Paolo P. Lava_, Apr 02 2019

%t nmax = 52; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 52; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 1) x^k/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

%Y Cf. A000005, A000203, A032741, A081362, A107742, A318783.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Sep 04 2018

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Last modified March 29 08:01 EDT 2024. Contains 371265 sequences. (Running on oeis4.)