A308564
Expansion of e.g.f. Sum_{k>=1} phi(k)*(exp(x) - 1)^k/k!, where phi = Euler totient function (A000010).
Original entry on oeis.org
1, 2, 6, 22, 90, 404, 1974, 10366, 57864, 341690, 2134022, 14104624, 98498972, 723664482, 5561589508, 44473028634, 368602225688, 3159852790392, 27997141025686, 256410638073082, 2428063270357748, 23774001479212114, 240580239864321604, 2513553050765310236
Offset: 1
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b:= proc(n, m) option remember; uses numtheory;
`if`(n=0, phi(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=1..24); # Alois P. Heinz, Aug 04 2021
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nmax = 24; Rest[CoefficientList[Series[Sum[EulerPhi[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[EulerPhi[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] EulerPhi[k], {k, 1, n}], {n, 1, 24}]
A324913
a(n) = Sum_{k=1..n} 2^k * phi(k), where phi is the Euler totient function A000010.
Original entry on oeis.org
2, 6, 22, 54, 182, 310, 1078, 2102, 5174, 9270, 29750, 46134, 144438, 242742, 504886, 1029174, 3126326, 4699190, 14136374, 22524982, 47690806, 89633846, 274183222, 408400950, 1079489590, 1884795958, 4300715062, 7521940534, 22554326070, 31144260662, 95568770102
Offset: 1
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ListTools:-PartialSums([seq(2^k * numtheory:-phi(k),k=1..50)]); # Robert Israel, Apr 17 2025
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Accumulate[Table[2^k*EulerPhi[k], {k, 1, 40}]]
A356340
a(n) = Sum_{k=1..n} binomial(2*n, n-k) * phi(k), where phi is the Euler totient function.
Original entry on oeis.org
1, 5, 23, 102, 444, 1909, 8133, 34404, 144714, 605920, 2527348, 10507978, 43569096, 180219699, 743907057, 3065019864, 12607648238, 51783970314, 212412697368, 870249992168, 3561502879100, 14560944187796, 59476980459794, 242741090637012, 989921853052930, 4034101567907172
Offset: 1
-
Table[Sum[Binomial[2*n, n-k]*EulerPhi[k], {k, 1, n}], {n, 1, 30}]
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a(n) = sum(k=1, n, binomial(2*n, n-k) * eulerphi(k)); \\ Michel Marcus, Aug 05 2022
A330148
a(n) = Sum_{k=1..n} binomial(n,k) * phi(k) * phi(n - k + 1), where phi = A000010.
Original entry on oeis.org
1, 3, 11, 30, 94, 238, 692, 1596, 4536, 9350, 27840, 52884, 149668, 294838, 782432, 1463224, 4095792, 7460274, 20229356, 36847380, 100317284, 170262974, 492659240, 814679680, 2184447760, 3965791284, 9988168320, 17883230712, 49362800340, 80674575956, 213420581248
Offset: 1
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[&+[Binomial(n,k)*EulerPhi(k)*EulerPhi(n-k+1):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Dec 03 2019
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Table[Sum[Binomial[n, k] EulerPhi[k] EulerPhi[n - k + 1], {k, 1, n}], {n, 1, 31}]
nmax = 31; CoefficientList[Series[(1/2) D[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n) = sum(k=1, n, binomial(n,k)*eulerphi(k)*eulerphi(n-k+1)); \\ Michel Marcus, Dec 03 2019
A356343
a(n) = Sum_{k=1..n} binomial(2*n, k) * phi(k), where phi is the Euler totient function.
Original entry on oeis.org
2, 10, 61, 288, 1723, 6524, 37441, 158504, 737019, 2867500, 15200293, 56951428, 291648771, 1141099348, 4686310739, 19016248192, 95307214595, 358297247772, 1748879020425, 6725041736572, 27649247188159, 108460437728204, 522912325647543, 1966622896068784, 8831400010510925
Offset: 1
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Table[Sum[Binomial[2*n, k]*EulerPhi[k], {k, 1, n}], {n, 1, 30}]
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a(n) = sum(k=1, n, binomial(2*n, k) * eulerphi(k)); \\ Michel Marcus, Aug 05 2022
A356346
a(n) = Sum_{k=1..n} binomial(2*k, k) * phi(k), where phi is the Euler totient function.
Original entry on oeis.org
2, 8, 48, 188, 1196, 3044, 23636, 75116, 366836, 1105860, 8160180, 18976804, 143784004, 384483604, 1625423764, 6434066884, 43771766404, 98222578204, 734437326604, 1837209557164, 8296304050444, 29337293687644, 210472769694844, 468453599159644, 2996665727914684
Offset: 1
-
Table[Sum[Binomial[2*k, k]*EulerPhi[k], {k, 1, n}], {n, 1, 30}]
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a(n) = sum(k=1, n, binomial(2*k, k) * eulerphi(k)); \\ Michel Marcus, Aug 05 2022
Showing 1-6 of 6 results.
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