A280541 -id:A280541 - cp's OEIS frontend

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

1, 1, 5, 11, 33, 67, 180, 366, 871, 1782, 3927, 7885, 16637, 32763, 66469, 128938, 253871, 484034, 930959, 1747304, 3292730, 6092664, 11282364, 20596790, 37568653, 67736175, 121886533, 217261372, 386216073, 681119439, 1197524035, 2091091902, 3639519280

Offset: 0

Vaclav Kotesovec, Jan 05 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A006171, A038040, A061256, A107742, A192065, A280541.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[k*DivisorSigma[0, k], j]*(-1)^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 - x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

log(a(n)) ~ (3/2)^(2/3) * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3). - Vaclav Kotesovec, Aug 28 2018

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

Original entry on oeis.org

1, 1, 6, 15, 48, 108, 323, 716, 1868, 4217, 10137, 22311, 51477, 110817, 245260, 519918, 1114914, 2318557, 4854952, 9923533, 20335761, 40941170, 82365742, 163413699, 323589060, 633429923, 1236392498, 2390718266, 4606489839, 8805346615, 16768968317, 31713677061, 59747953446

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A034718, A280473, A280541, A318413.

a:=series(mul(mul(mul((1+x^(i*j*k))^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(i j k), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}]  x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
nmax = 32; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A034718[[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}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 31 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*tau_3(k)), where tau_3() = A007425.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
Conjecture: log(a(n)) ~ 3^(2/3) * Zeta(3)^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3). - Vaclav Kotesovec, Sep 02 2018

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

Original entry on oeis.org

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).

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

Original entry on oeis.org

1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

seq(n!*coeff(series(mul((1+x^k)^(tau(k)/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))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) 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[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]/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}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2018 *)

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

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

A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 8, 11, 15, 24, 34, 43, 63, 87, 115, 159, 217, 279, 380, 505, 657, 868, 1139, 1458, 1913, 2482, 3162, 4069, 5232, 6628, 8469, 10755, 13544, 17127, 21634, 27061, 33988, 42557, 52985, 66069, 82289, 101862, 126281, 156275, 192655, 237530, 292502

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327064, A327065, A327066.

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

A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

Original entry on oeis.org

1, 1, 2, 5, 10, 18, 35, 62, 110, 197, 339, 573, 975, 1621, 2674, 4385, 7108, 11422, 18277, 28976, 45648, 71531, 111372, 172416, 265695, 407210, 621143, 943392, 1426414, 2147672, 3221271, 4812534, 7163440, 10625651, 15706871, 23141148, 33987287, 49762235

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327065, A327067.

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

A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).

Original entry on oeis.org

1, 1, 2, 5, 12, 20, 42, 75, 141, 259, 466, 799, 1427, 2443, 4169, 7049, 11863, 19605, 32518, 53184, 86579, 140018, 225380, 359739, 572864, 905903, 1426270, 2234952, 3488313, 5416403, 8383226, 12917257, 19831763, 30334937, 46245977, 70242043, 106371686

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327064, A327068.

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

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

Original entry on oeis.org

1, 2, 10, 30, 98, 270, 786, 2046, 5418, 13556, 33726, 81002, 192902, 447562, 1027990, 2316750, 5165398, 11345298, 24668952, 52972902, 112688802, 237193354, 494933514, 1023238806, 2098662698, 4269141516, 8620916966, 17280687472, 34405835066, 68044209950, 133732805458

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

nonn

Convolution of A280540 and A280541.

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

Cf. A000005, A038040, A156616, A280540, A280541, A301554, A301555.

a:=series(mul(mul(((1+x^(i*j))/(1-x^(i*j)))^(i*j),j=1..100),i=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019

nmax = 30; CoefficientList[Series[Product[Product[((1 + x^(i j))/(1 - x^(i j)))^(i j), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k DivisorSigma[0, k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(1 - (-1)^(k/d)) d^2 DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*tau(k)), where tau(k) = number of divisors of k (A000005).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (1 - (-1)^(k/d))*d^2*tau(d) ) * x^k/k).
log(a(n)) ~ 3^(2/3) * (7*Zeta(3))^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Sep 03 2018

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]

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).

Original entry on oeis.org

1, 1, 5, 15, 53, 165, 561, 1807, 5993, 19586, 64491, 211466, 695101, 2281614, 7494995, 24610588, 80829373, 265437828, 871738976, 2862815763, 9401768055, 30875971366, 101399191222, 333001988025, 1093603789613, 3591473940515, 11794667169894, 38734550365835, 127207121681103, 417757532953031

Offset: 0

Ilya Gutkovskiy, Aug 27 2018

nonn

Cf. A000005, A038040, A088305, A129921, A180305, A280540, A280541.

a:=series(1/(1-add(add(i*j*x^(i*j),j=1..100),i=1..100)),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} k*x^k/(1 - x^k)^2).
G.f.: 1/(1 - Sum_{k>=1} k*d(k)*x^k), where d(k) = number of divisors of k (A000005).
a(0) = 1; a(n) = Sum_{k=1..n} A038040(k)*a(n-k).
a(n) ~ c / r^n, where r = 0.304499876501217750838861744045680232405337905509126... is the root of the equation Sum_{k>=1} k*r^k/(1 - r^k)^2 = 1 and c = 0.44152042515136849968144466258954953693306684400261343177792428746297872748... - Vaclav Kotesovec, Aug 28 2018

cp's OEIS Frontend

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(i*j).

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A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(i*j*k).

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

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A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).

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A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

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A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

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A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^(k*j)).

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A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).

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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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A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

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

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

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

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).