A318416 -id:A318416 - cp's OEIS frontend

A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - ijx^(i*j)).

1, 1, 5, 11, 35, 69, 200, 398, 1014, 2069, 4820, 9716, 21787, 43209, 92530, 182773, 378676, 737526, 1492451, 2872788, 5686194, 10837935, 21052463, 39699970, 75972300, 141818166, 267607065, 495142606, 922920753, 1692529453, 3121105278, 5676677651, 10364752129, 18708292447, 33851433117, 60656841965

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

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

Cf. A000005, A006171, A006906, A280540, A318416.

nmax = 35; CoefficientList[Series[Product[Product[1/(1 - i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Product[1/(1 - k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[Sum[d^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*(-1)^j*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=1} 1/(1 - k*x^k)^tau(k), where tau = number of divisors (A000005).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^(k/d+1)*tau(d) ) * x^k/k).
From Vaclav Kotesovec, Aug 27 2018: (Start)
a(n) ~ c * n * 3^(n/3), where
c = 10751825728554.298582954430359167227238488440778317... if mod(n,3)=0
c = 10751825728553.835664124121831524829543267756895348... if mod(n,3)=1
c = 10751825728553.838520991588115910603754564083195806... if mod(n,3)=2
In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(0,k)))) / (21 - 16*3^(1/3) + 3^(2/3)) - (3*Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(0,k)))) / ((-1)^(2*n/3)*((3 + 2*(-3)^(1/3))^2*(-3 + (-3)^(2/3)))) + Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(0,k))) / (9*(-1)^(4*n/3)*((1 + (-1/3)^(1/3))*(1 - 2*(-1/3)^(2/3))^2))
(End)

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).

A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 6, 18, 52, 142, 404, 1018, 2624, 6645, 16124, 38857, 92245, 214841, 494098, 1125062, 2522188, 5604930, 12327860, 26838595, 57913194, 123951482, 263019720, 553989989, 1158449522, 2405179547, 4961047246, 10168544537, 20714279168, 41952595411, 84494479578

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

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

Cf. A000203, A022629, A266891, A318416, A107742, A192065, A318483.

nmax = 40; CoefficientList[Series[Product[(1+k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x]

A333653 Expansion of Product_{i>=1, j>=1} (1 + ix^(ij)).

Original entry on oeis.org

1, 1, 3, 7, 13, 27, 54, 98, 174, 335, 572, 1004, 1733, 2933, 4916, 8307, 13470, 22042, 35851, 57256, 91462, 145231, 227667, 355522, 554058, 853986, 1313121, 2010318, 3057827, 4627213, 6989808, 10481205, 15679549, 23365207, 34658909, 51241077, 75541695, 110852295, 162238415

Offset: 0

Seiichi Manyama, Aug 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A022629, A107742, A285244, A318416, A332199.

m = 38; CoefficientList[Series[Product[1 + i*x^(i*j), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)

N=40; x='x+O('x^N); Vec(prod(i=1, N, prod(j=1, N\i, 1+i*x^(i*j))))

N=40; x='x+O('x^N); Vec(prod(k=1, N, prod(d=1, k, 1+(k%d==0)*d*x^k)))

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} (1 + i*q^i).

G.f.: Product_{k>0} Product_{d|k} (1 + d*x^k).

A318482 Expansion of Product_{i>=1, j>=1, k>=1} (1 + ijk*x^(ijk)).

Original entry on oeis.org

1, 1, 6, 15, 45, 105, 302, 668, 1664, 3830, 8793, 19350, 43265, 92552, 198418, 418128, 869999, 1787964, 3651028, 7353349, 14697367, 29139447, 57225893, 111572329, 216001937, 415000057, 792008753, 1502559866, 2831393559, 5305614223, 9885825732, 18318165218

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Cf. A022629, A318416.

Cf. A000009, A280541, A318414, A318481.

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

cp's OEIS Frontend

A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - i*j*x^(i*j)).

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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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A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

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Mathematica

A333653 Expansion of Product_{i>=1, j>=1} (1 + i*x^(i*j)).

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PARI

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A318482 Expansion of Product_{i>=1, j>=1, k>=1} (1 + i*j*k*x^(i*j*k)).

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Mathematica

A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - ijx^(i*j)).

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

A333653 Expansion of Product_{i>=1, j>=1} (1 + ix^(ij)).

A318482 Expansion of Product_{i>=1, j>=1, k>=1} (1 + ijk*x^(ijk)).