A107742 -id:A107742 - cp's OEIS frontend

A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

1, 1, 513, 20197, 413669, 12445003, 372981573, 9158438541, 223776496101, 5567873958982, 132009631562091, 3018411978731059, 68171158091244082, 1512439928316217508, 32796174722883608382, 698503712498547606328, 14656105328324700415778, 302787437988353941515934

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..995

Cf. A107742 (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

Cf. A013957, A301547.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(11 * Pi^(10/11) * (31*Zeta(11))^(1/11) * n^(10/11) / (2^(13/11) * 5^(10/11))) * (155*Zeta(11)/Pi)^(1/22) / (2^(155/264) * sqrt(11) * n^(6/11)).

G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A320235 G.f.: Product_{k>=1, j>=1} (1 + x^(k*j))^2.

Original entry on oeis.org

1, 2, 5, 12, 24, 48, 94, 172, 310, 550, 946, 1602, 2679, 4394, 7123, 11424, 18082, 28344, 44039, 67754, 103412, 156660, 235489, 351602, 521650, 768998, 1127100, 1642946, 2381929, 3436028, 4932998, 7049004, 10028422, 14207122, 20044327, 28169528, 39439899

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Self-convolution of A107742.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742, A280451, A320236.

nmax = 50; CoefficientList[Series[Product[(1+x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/3).

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 89, 115, 152, 204, 260, 336, 438, 552, 702, 896, 1117, 1400, 1758, 2171, 2688, 3332, 4079, 5000, 6131, 7446, 9048, 10992, 13255, 15984, 19264, 23081, 27644, 33084, 39408, 46912, 55797, 66107, 78264, 92572, 109140

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327046, A327047.

Cf. A107742, A327042.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 11^(1/4) * exp(sqrt(11*n/2)*Pi/3) / (2^(13/4)*sqrt(3)*n^(3/4)).

A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 30, 45, 62, 85, 120, 161, 216, 293, 385, 505, 667, 862, 1112, 1438, 1833, 2330, 2965, 3733, 4688, 5887, 7334, 9114, 11319, 13970, 17203, 21162, 25905, 31643, 38605, 46911, 56891, 68904, 83179, 100224, 120603, 144719, 173360, 207396

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327045, A327047.

Cf. A107742, A327043.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(5) * exp(5*Pi*sqrt(n)/6) / (16*sqrt(3)*n^(3/4)).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 23, 34, 51, 72, 101, 143, 195, 267, 366, 487, 650, 866, 1135, 1487, 1940, 2504, 3226, 4145, 5283, 6714, 8513, 10725, 13481, 16905, 21085, 26244, 32588, 40299, 49732, 61229, 75131, 92004, 112435, 137009, 166627, 202269, 244919, 296038

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} (1 + x^(j*k))), then a(n) ~ HarmonicNumber(m)^(1/4) * exp(Pi*sqrt(HarmonicNumber(m)*n/3)) / (2^((m+3)/2) * 3^(1/4) * n^(3/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327045, A327046.

Cf. A107742, A327044.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)) * (1+x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 137^(1/4) * exp(sqrt(137*n/5)*Pi/6) / (2^(9/2)*sqrt(3)*5^(1/4)*n^(3/4)).

A280663 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^3)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 32, 39, 47, 57, 68, 81, 97, 115, 136, 162, 190, 223, 263, 306, 357, 417, 483, 561, 650, 750, 866, 997, 1145, 1315, 1507, 1725, 1971, 2250, 2564, 2917, 3318, 3766, 4270, 4840, 5475, 6188, 6990, 7881, 8881

Offset: 0

Vaclav Kotesovec, Jan 07 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742, A280451, A280664.

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

a(n) ~ exp(Pi*sqrt(Zeta(3)*n/3) + (2^(1/3)-1) * Pi^(-1/3) * Gamma(4/3) * Zeta(4/3) * Zeta(1/3) * (3*n/Zeta(3))^(1/6)) * Zeta(3)^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).

A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij)/(ij)).

Original entry on oeis.org

1, 1, 2, 10, 40, 248, 1868, 14516, 131920, 1409040, 15697872, 191687472, 2663239104, 37878672960, 582357866400, 9898540886880, 172534018584960, 3192686545714560, 63844374067107840, 1309775114921541120, 28512040933544970240, 656888836504576112640, 15495311684125737031680

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..447

Cf. A000005, A007838, A107742, A181541, A318416, A318693.

seq(n!*coeff(series(mul((1+x^k/k)^tau(k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

E.g.f.: Product_{k>=1} (1 + x^k/k)^tau(k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-d)^(1-k/d)*tau(d) ) * x^k/k).

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

A320238 G.f.: Product_{k>=1, j>=1} (1 + x^(kj)) / (1 - x^(kj))^2.

Original entry on oeis.org

1, 3, 11, 31, 85, 209, 504, 1138, 2514, 5339, 11098, 22432, 44535, 86523, 165496, 311187, 577190, 1055524, 1907423, 3405574, 6016826, 10520065, 18222215, 31275320, 53230224, 89860112, 150551503, 250388180, 413572707, 678574627, 1106396434, 1793009335

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Convolution of A107742 and A320236.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A158441, A107742, A320236.

nmax = 50; CoefficientList[Series[Product[(1+x^(k*j))/(1-x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ Pi * sqrt(5*n*log(n)/6).

A321192 a(n) = [x^n] Product_{k>=1} (1 + x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

Original entry on oeis.org

1, 1, 2, 6, 20, 55, 239, 700, 3212, 10104, 48622, 161579, 806843, 2799199, 14379647, 52018828, 273472712, 1023655306, 5491615463, 21234676241, 115910309103, 460998296937, 2556361045845, 10440651927427, 58714921974979, 245586789818255, 1399187406060485

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000009, A077592, A107742, A280473, A280486, A321191.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 30; Table[SeriesCoefficient[Product[(1 + x^k)^tau[k, n], {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

a(n) = [x^n] Product_{k_1>=1, k_2>=1, ..., k_n>=1} (1 + x^(k_1*k_2*...*k_n)).

cp's OEIS Frontend

A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

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A320235 G.f.: Product_{k>=1, j>=1} (1 + x^(k*j))^2.

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A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)).

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A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)).

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A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*k)).

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A280663 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^3)).

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A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).

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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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A320238 G.f.: Product_{k>=1, j>=1} (1 + x^(k*j)) / (1 - x^(k*j))^2.

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A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*k)).

A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij)/(ij)).

A320238 G.f.: Product_{k>=1, j>=1} (1 + x^(kj)) / (1 - x^(kj))^2.