A318415 -id:A318415 - cp's OEIS frontend

A318416 Expansion of Product_{i>=1, j>=1} (1 + ijx^(i*j)).

1, 1, 4, 10, 22, 50, 115, 231, 470, 995, 1912, 3745, 7222, 13608, 25345, 47322, 85654, 155163, 278867, 494080, 870618, 1524769, 2640527, 4549564, 7802037, 13251684, 22412317, 37706268, 63015263, 104800015, 173574936, 285694401, 468449681, 764775169, 1242535747, 2010866469, 3242127656

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

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

Cf. A000005, A022629, A107742, A280541, A318415.

a:=series(mul(mul(1+i*j*x^(i*j),j=1..55),i=1..55),x=0,37): seq(coeff(a,x,n),n=0..36); # Paolo P. Lava, Apr 02 2019

nmax = 36; CoefficientList[Series[Product[Product[(1 + i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Product[(1 + k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; 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, 36}]
nmax = 36; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, 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] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=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).

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

Original entry on oeis.org

1, 1, 7, 16, 64, 133, 465, 1008, 3023, 6695, 18206, 40175, 103229, 225470, 549873, 1194620, 2801742, 6015042, 13686306, 29063919, 64424496, 135362432, 293512852, 610061141, 1298516539, 2670738781, 5591712472, 11388116508, 23499720744, 47403692965, 96564236754

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A006906, A318415.

Cf. A000041, A280540, A318413, A318482.

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

A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 7, 19, 71, 173, 583, 1443, 4255, 10648, 28929, 71159, 184740, 445626, 1110122, 2638328, 6369490, 14870194, 35031627, 80465028, 185556696, 419916149, 950785580, 2121471778, 4727971847, 10412230698, 22876886529, 49776871862, 107974178843, 232302695301

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5920

Cf. A000203, A006906, A266941, A318415, A006171, A061256, A318484.

nmax = 40; CoefficientList[Series[Product[1/(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]*(-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]

a(n) ~ c * n^3 * 3^(n/3), where
c = 280631952508395331283883354935233682635.581151020... if mod(n,3)=0
c = 280631952508395331283883354935233682635.059082354... if mod(n,3)=1
c = 280631952508395331283883354935233682635.088610121... if mod(n,3)=2
In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(k)))/(18*(57 - 90*3^(1/3) + 35*3^(2/3)))) - Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(k)))/ ((-1)^(2*n/3)*(6*(3 + 2*(-3)^(1/3))^3*(-3 + (-3)^(2/3)))) - ((-1)^(1 - (4*n)/3)*Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(k))))/(486*(1 + (-1/3)^(1/3))* (1 - 2*(-1/3)^(2/3))^3)

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij)/(ij)).

Original entry on oeis.org

1, 1, 4, 16, 100, 628, 5388, 46212, 491328, 5381760, 68023056, 892073136, 13238778144, 201822014496, 3397195558560, 59356290115296, 1121097742183296, 21916440531679104, 459855848691876096, 9952944631606759680, 229191463614349301760, 5446997871156332605440, 136439919208493792455680

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Cf. A000005, A006171, A007841, A294469, A318415, A318694.

seq(n!*coeff(series(mul(mul(1/(1-x^(i*j)/(i*j)),i=1..30),j=1..30),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[1/(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/(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/(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).

A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - ix^(ij)).

Original entry on oeis.org

1, 1, 4, 8, 22, 40, 101, 183, 412, 765, 1586, 2899, 5834, 10484, 20199, 36246, 67758, 119837, 219661, 384200, 690164, 1197423, 2114105, 3632088, 6332797, 10779478, 18555115, 31354932, 53385037, 89494901, 150983344, 251284829, 420218575, 694947117, 1152915743, 1894656801

Offset: 0

Seiichi Manyama, Aug 23 2020

nonn

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

Cf. A006171, A006906, A285245, A318415, A333653.

m = 35; CoefficientList[Series[Product[1/(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(1/prod(i=1, N, prod(j=1, N\i, 1-i*x^(i*j))))

N=40; x='x+O('x^N); Vec(1/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/(1 - i*q^i).

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

cp's OEIS Frontend

A318416 Expansion of Product_{i>=1, j>=1} (1 + i*j*x^(i*j)).

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A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - i*j*k*x^(i*j*k)).

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A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

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A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j)/(i*j)).

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A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - i*x^(i*j)).

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A318416 Expansion of Product_{i>=1, j>=1} (1 + ijx^(i*j)).

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij)/(ij)).

A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - ix^(ij)).