A301511
Expansion of e.g.f. exp(Sum_{k>=1} psi(k)*x^k/k!), where psi() is the Dedekind psi function (A001615).
Original entry on oeis.org
1, 1, 4, 14, 68, 362, 2224, 14940, 110348, 878600, 7518002, 68529122, 662709832, 6764329158, 72622813172, 817239648500, 9612724174088, 117878757097178, 1503660164683864, 19911519090176808, 273221610513382028, 3878513600608651636, 56873187579428449852, 860296560100458300892
Offset: 0
E.g.f.: A(x) = 1 + x/1! + 4*x^2/2! + 14*x^3/3! + 68*x^4/4! + 362*x^5/5! + 2224*x^6/6! + 14940*x^7/7! + ...
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psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; a[n_] := a[n] = SeriesCoefficient[Exp[Sum[psi[k] x^k/k!, {k, 1, n}]], {x, 0, n}]; Table[a[n] n!, {n, 0, 23}]
psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; a[n_] := a[n] = Sum[psi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
A328053
Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).
Original entry on oeis.org
0, 1, 0, 1, -3, 8, -32, 166, -926, 5842, -42812, 348632, -3088388, 29871372, -314102574, 3554714938, -43057252520, 556487433400, -7644034688586, 111160926400032, -1706191876272876, 27567942738717360, -467712309003533398, 8312805777830133096
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
*binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[phi]))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 06 2019
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nmax = 23; CoefficientList[Series[Log[1 + Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = EulerPhi[n] - Sum[Binomial[n, k] EulerPhi[n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]
A352887
Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k)*x^k/k!), where phi is the Euler totient function A000010.
Original entry on oeis.org
1, 1, 3, 14, 84, 634, 5740, 60626, 731852, 9938670, 149966116, 2489148386, 45070961740, 884107377360, 18676602726734, 422721143355808, 10205605681874952, 261789688633794528, 7110331886095458918, 203848868169846041430, 6151813078359073154568
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k!))))
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a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n, k)*a(n-k)));
Showing 1-3 of 3 results.
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