A307241 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*d(k+1)*a(n-k), where d() is the number of divisors (A000005).
1, 2, 2, 3, 6, 12, 23, 42, 75, 135, 248, 460, 849, 1554, 2837, 5192, 9527, 17490, 32083, 58809, 107781, 197578, 362280, 664320, 1218069, 2233202, 4094289, 7506602, 13763219, 25234674, 46266927, 84828138, 155528132, 285154061, 522819002, 958568628, 1757496665, 3222295912
Offset: 0
Keywords
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[0, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}] nmax = 37; CoefficientList[Series[-x/Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x] nmax = 37; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^(1/k), {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]
Formula
G.f.: -x / Sum_{k>=1} (-x)^k/(1 - (-x)^k).
G.f.: 1 / (d/dx) log(Product_{k>=1} (1 - (-x)^k)^(1/k)).