A032308 -id:A032308 - cp's OEIS frontend

A343465 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-3)^d.

3, -3, 11, -21, 51, -119, 315, -831, 2195, -5883, 16107, -44357, 122643, -341487, 956635, -2690841, 7596483, -21522347, 61171659, -174342165, 498112275, -1426403751, 4093181691, -11767920107, 33891544419, -97764009003, 282429537947, -817028472645, 2366564736723, -6863037262207

Offset: 1

Ilya Gutkovskiy, Apr 16 2021

sign

Cf. A000010, A001867, A032308, A038064, A074763, A261582, A343466, A343467.

Table[-(1/n) Sum[EulerPhi[n/d] (-3)^d, {d, Divisors[n]}], {n, 1, 30}]
nmax = 30; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 3 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} phi(k) * log(1 + 3*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-3)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A032308.
Product_{n>=1} (1 - x^n)^a(n) = g.f. for A261582.

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts.

A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

Original entry on oeis.org

1, 6, 6, 348, -570, 12084, -31332, 780792, -6111930, 65506884, -599418444, 6707736456, -69508986852, 738378468744, -7878832564872, 85524000547056, -929068361832378, 10158667075255524, -111690827626777788, 1234592278534799592, -13700571880245603276, 152613494540593338264

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032308, A370714.

Cf. A300579, A344062.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-3, x]/4], {x, 0, nmax}], x] * 4^Range[0, nmax]

G.f.: Product_{k>=1} (1 + 3*(4*x)^k)^(1/2).

a(n) ~ (-1)^(n+1) * c * 12^n / n^(3/2), where c = QPochhammer(-1/3)^(1/2) / (2*sqrt(Pi)) = 0.311283382185276347775502154581850436407169685238...

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

Original entry on oeis.org

1, 3, 0, 99, -270, 2430, -10287, 105462, -750141, 5702481, -42623901, 347424633, -2779077762, 22353287634, -181730796723, 1493711042589, -12321529794261, 102125312638713, -850797139405887, 7120067746384863, -59800770201017934, 503922807927384129, -4259721779079782751

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032308, A370710.

Cf. A300579, A344062.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

G.f.: Product_{k>=1} (1 + 3*(3*x)^k)^(1/3).

a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...

A266820 Expansion of Product_{k>=1} ((1 + 2x^k) (1 + 3*x^k)).

Original entry on oeis.org

1, 5, 11, 30, 66, 115, 252, 445, 762, 1350, 2238, 3690, 5909, 9480, 14460, 22475, 34326, 51150, 76398, 111810, 163350, 236610, 339667, 482040, 684060, 960780, 1340953, 1863570, 2573022, 3533310, 4830822, 6580170, 8900382, 12011430, 16125198, 21567965

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A032302 and A032308.

In general, for m1 > 0 and m2 > 0, if g.f. = Product_{k>=1} ((1 + m1*x^k) * (1 + m2*x^k)) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m1+1)*(m2+1)*Pi) * n^(3/4)), where c = Pi^2/3 + log(m1)^2/2 + log(m2)^2/2 + polylog(2, -1/m1) + polylog(2, -1/m2).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A032302, A032308, A266819, A266822.

nmax = 40; CoefficientList[Series[Product[(1+2*x^k) * (1+3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi) * n^(3/4)), where c = Pi^2/3 + log(2)^2/2 + log(3)^2/2 + polylog(2, -1/2) + polylog(2, -1/3) = 6.665989921346842772385004076363525173910446415877... .

A266822 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 3*x^k)).

Original entry on oeis.org

1, 4, 7, 20, 35, 60, 124, 200, 324, 524, 865, 1320, 2016, 3036, 4453, 6684, 9668, 13856, 19792, 27876, 38956, 54640, 75320, 103268, 141191, 191320, 257892, 346164, 463284, 615292, 814883, 1074556, 1409904, 1844284, 2402756, 3118020, 4038164, 5207344, 6694116

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A000009 and A032308.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A032308, A266819, A266820.

nmax = 40; CoefficientList[Series[Product[(1+x^k) * (1+3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(2*Pi) * n^(3/4)), where c = Pi^2/4 + log(3)^2/2 + polylog(2, -1/3) = 2.761842454190822171313479302500904035832... .

A268499 Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 2, 0, 8, -2, 8, 16, 8, 8, 10, 80, -8, 72, -24, 144, 128, 134, 40, 224, 120, 232, 688, 176, 696, 32, 1194, -96, 1840, 1144, 2248, 288, 2968, 800, 4160, 752, 5104, 6438, 4984, 5104, 5488, 10960, 4856, 14080, 3480, 24408, 15448, 26832, 7080, 42120, 11178

Offset: 0

Vaclav Kotesovec, Feb 06 2016

sign

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 + x^k)) then a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt((m+1)*Pi) * n^(3/4)), where c = Pi^2/3 + 2*log(m)^2 + 4*polylog(2, -1/m).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A032308, A266821, A268498, A268500.

nmax = 100; CoefficientList[Series[Product[(1+3*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(3)^2 + 4*polylog(2, -1/3) = 4.467633549370382939364... .

A291971 Triangle read by rows: T(n,k) = 3 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 3, 0, 3, 0, 3, 9, 0, 3, 9, 0, 3, 18, 0, 3, 18, 27, 0, 3, 27, 27, 0, 3, 27, 54, 0, 3, 36, 81, 0, 3, 36, 108, 81, 0, 3, 45, 135, 81, 0, 3, 45, 189, 162, 0, 3, 54, 216, 243, 0, 3, 54, 270, 405, 0, 3, 63, 324, 486, 243, 0, 3, 63, 378, 729, 243, 0, 3, 72, 432, 891

Offset: 0

Seiichi Manyama, Sep 07 2017

			First few rows are:
  1;
  0, 3;
  0, 3;
  0, 3,  9;
  0, 3,  9;
  0, 3, 18;
  0, 3, 18,  27;
  0, 3, 27,  27;
  0, 3, 27,  54;
  0, 3, 36,  81;
  0, 3, 36, 108, 81.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A032308.
Columns 0-1 give A000007, A010701.
Cf. A008289 (m=1), A291970 (m=2), this sequence (m=3).

A370739 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, -75, 35250, -1138125, 72645000, -3307996875, 244578890625, -15502648125000, 985908809765625, -63515254624218750, 4314500023927734375, -291905297026816406250, 19789483493484814453125, -1355414138248614990234375, 93666904586649390380859375, -6498800175020013123779296875

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

In general, if d > 1, m > 1 and g.f. = Product_{k>=1} (1 + d*x^k)^(1/m), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d)^(1/m) * d^n / (m*Gamma(1 - 1/m) * n^(1 + 1/m)).

Cf. A032308 (d=3,m=1), A370711 (d=3,m=2), A370712 (d=3,m=3), A370738 (d=3,m=4).
Cf. A032302 (d=2,m=1), A370709 (d=2,m=2), A370716 (d=2,m=3), A370736 (d=2,m=4), A370737 (d=2,m=5).
Cf. A000009 (d=1,m=1), A298994 (d=1,m=2), A303074 (d=1,m=3)

nmax = 20; CoefficientList[Series[Product[1+3*x^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+3*(25*x)^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + 3*(25*x)^k)^(1/5).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/5) * 75^n / (5 * Gamma(4/5) * n^(6/5)).

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).

Original entry on oeis.org

1, 6, 36, 360, 1998, 18792, 121176, 1123632, 7537860, 72078174, 510702408, 4896308088, 35923749480, 345406994280, 2600934294816, 24985346997888, 191735328374478, 1838307293836560, 14317601666954364, 136953233511162840, 1079293961918593800, 10299943344889922832

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} ((1 + d*x^k)/(1 - d*x^k))^(1/m), then a(n) ~ QPochhammer(-1, 1/d)^(1/m) * d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Cf. A303390 (d=3,m=1), A370751 (d=3,m=2), A370752 (d=3,m=3).
Cf. A261584 (d=2,m=1), A303346 (d=2,m=2), A370750 (d=2,m=3), A370749 (d=2,m=4).
Cf. A015128 (d=1,m=1), A303307 (d=1,m=2), A303342 (d=1,m=3).
Cf. A032308, A242587.
Cf. A370712, A370710.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k)/(1 - 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k)/(1 - 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + 3*(3*x)^k)/(1 - 3*(3*x)^k))^(1/3).

a(n) ~ QPochhammer(-1, 1/3)^(1/3) * 9^n / (Gamma(1/3) * QPochhammer(1/3)^(1/3) * n^(2/3)).

cp's OEIS Frontend

A343465 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-3)^d.

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A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

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A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

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A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

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A266820 Expansion of Product_{k>=1} ((1 + 2*x^k) * (1 + 3*x^k)).

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A266822 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 3*x^k)).

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A268499 Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).

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A291971 Triangle read by rows: T(n,k) = 3 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A370739 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

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A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/3).

A266820 Expansion of Product_{k>=1} ((1 + 2x^k) (1 + 3*x^k)).

A370739 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).