A280540 -id:A280540 - cp's OEIS frontend

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).

1, 1, 4, 10, 24, 52, 125, 253, 549, 1126, 2290, 4525, 8987, 17259, 33174, 62669, 117425, 217295, 399904, 726984, 1314257, 2354807, 4191671, 7405590, 13009916, 22696115, 39384232, 67937488, 116584833, 199001304, 338076500, 571507377, 961855945, 1611567819

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, A280540.

nmax = 50; CoefficientList[Series[Product[(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]*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 + x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

Conjecture: log(a(n)) ~ 3 * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Aug 29 2018

A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(ijk).

Original entry on oeis.org

1, 1, 7, 16, 61, 130, 429, 945, 2684, 5990, 15530, 34313, 83995, 183070, 427046, 919480, 2067589, 4384678, 9577536, 20019243, 42664087, 87954522, 183573639, 373430131, 765524808, 1537737243, 3102614407, 6159028445, 12252086879, 24051526041, 47239506797, 91765428710, 178156003047

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, A174465, A174467, A280540, A318414.

a:=series(mul(mul(mul(1/(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/(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/(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[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[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 = 50; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A034718[[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 31 2018 *)

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

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

Original entry on oeis.org

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)

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij))^(1/(ij)).

Original entry on oeis.org

1, 1, 4, 16, 106, 658, 6088, 51952, 592828, 6577948, 88213744, 1173121024, 18663391096, 289030343704, 5157010548064, 92428084599232, 1848308567352592, 37038307949425168, 822602470902709312, 18285742807660340992, 444405771941314880416, 10883864256927386369056, 286778106663948874858624

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
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, A007425, A028342, A280540, A305127, A318696, A318977.

seq(n!*coeff(series(mul(1/(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/(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/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[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[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)^(tau(k)/k), where tau = number of divisors (A000005).

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

A327066 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^j).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 17, 23, 41, 58, 93, 127, 205, 281, 423, 583, 869, 1180, 1716, 2322, 3317, 4479, 6282, 8406, 11696, 15589, 21343, 28325, 38480, 50756, 68307, 89688, 119725, 156586, 207449, 269921, 355530, 460804, 602816, 778281, 1012956, 1302481, 1686418

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327063, A327067, A327068.

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

A327067 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^k).

Original entry on oeis.org

1, 1, 3, 6, 15, 26, 57, 101, 202, 358, 670, 1165, 2113, 3614, 6326, 10691, 18275, 30408, 50969, 83716, 137943, 223883, 363547, 583369, 935524, 1485673, 2355496, 3705275, 5815497, 9066696, 14100325, 21802824, 33622951, 51592978, 78949673, 120278899, 182742752

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327064, A327066, A327068.

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

A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(kj))^(kj)).

Original entry on oeis.org

1, 1, 3, 6, 17, 28, 66, 116, 248, 441, 867, 1516, 2894, 5015, 9138, 15724, 27954, 47428, 82421, 138380, 235910, 392040, 657590, 1081225, 1789550, 2914500, 4763562, 7689071, 12433581, 19897139, 31862226, 50583981, 80285138, 126509709, 199167763, 311620226

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327065, A327066, A327067.

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

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]

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

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

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(i*j).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(i*j*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A327066 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^j).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A327067 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^(k*j)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).

A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(ijk).

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

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij))^(1/(ij)).

A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(kj))^(kj)).

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

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

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