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A318913
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Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k))^(1/prime(k)).
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5
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1, 0, 1, 2, 9, 44, 385, 1854, 18193, 153656, 1924641, 17123930, 276117721, 2880135972, 51150361249, 738748900694, 11608748988705, 198747251005424, 4029150617813953, 67937635488741426, 1607525018948543401, 32739373317847964060, 757174325538283357761, 16444280000832495199982
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OFFSET
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0,4
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LINKS
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FORMULA
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E.g.f.: exp(Sum_{k>=1} omega(k)*x^k/k), where omega(k) = number of distinct primes dividing k (A001221).
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MAPLE
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seq(n!*coeff(series(mul(1/(1-x^ithprime(k))^(1/ithprime(k)), k=1..100), x=0, 24), x, n), n=0..23); # Paolo P. Lava, Jan 09 2019
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MATHEMATICA
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nmax = 23; CoefficientList[Series[Product[1/(1 - x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[PrimeNu[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[PrimeNu[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
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PROG
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(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-isprime(k)*x^k)^(1/k)))) \\ Seiichi Manyama, Feb 28 2022
(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, omega(k)*x^k/k)))) \\ Seiichi Manyama, Feb 28 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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