A061256 -id:A061256 - cp's OEIS frontend

A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

1, 1, 6, 38, 526, 13074, 702813, 70939556, 13879861574, 5583837482767, 4393101918607162, 6717450870069292051, 21057681806321501744772, 131246096280071506595491449, 1604095619160115980216291007253, 40299198842857238408636666363954678, 2031474817845087309816967328335309651478

Offset: 0

Ilya Gutkovskiy, Oct 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547, A321042.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      sigma[k](d), d=divisors(j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

Table[SeriesCoefficient[Product[1/(1 - x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[1/(1 - x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

{a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^sigma(k, n)), n)} \\ Seiichi Manyama, Oct 27 2018

a(n) = [x^n] Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^n).

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^k))).

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

A301555 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma(k)).

Original entry on oeis.org

1, 2, 8, 22, 62, 154, 392, 914, 2136, 4776, 10544, 22626, 47982, 99538, 204100, 411714, 821130, 1616170, 3148812, 6066338, 11579954, 21893214, 41045780, 76306030, 140783060, 257789064, 468783092, 846697340, 1519599658, 2710476106, 4806507720, 8475250510

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Convolution of A061256 and A192065.

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000203, A061256, A192065.

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

a(n) ~ exp((3*Pi)^(2/3) * (7*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3) - 3^(1/3) * Pi^(4/3) * n^(1/3) / (2^(7/3) * (7*Zeta(3))^(1/3)) - 1/24 - Pi^2 / (224 * Zeta(3))) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(13/18) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^i. - Ilya Gutkovskiy, Aug 29 2018

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, -3, 4, -4, 4, -9, 14, -19, 30, -42, 50, -76, 128, -194, 286, -412, 598, -909, 1386, -2100, 3178, -4763, 7122, -10758, 16414, -25061, 38056, -57643, 87568, -133436, 203618, -311128, 475536, -726355, 1109718, -1697766, 2601166, -3987903, 6114666

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
seq(a(n), n = 0..43); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[EulerPhi,30]]

A301542 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_4(k)).

Original entry on oeis.org

1, 1, 18, 100, 526, 2546, 12953, 60929, 282194, 1265959, 5580958, 24057117, 101922204, 424244720, 1739362261, 7027590168, 28017627428, 110295521903, 429110693519, 1650961520518, 6285554480496, 23693047787961, 88469251486817, 327380976530282, 1201122749057307

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3315

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), this sequence (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A001159, A301548.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[4, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(3/2) * 3^(2/3) * Pi * (Zeta(5)/7)^(1/6) * n^(5/6)/5 + Pi * (7/Zeta(5))^(1/6) * n^(1/6) / (240 * sqrt(2) * 3^(2/3)) - 3*Zeta(5) / (8*Pi^4)) * Zeta(5)^(1/12) / (2^(3/4) * 3^(2/3) * 7^(1/12) * n^(7/12)).

G.f.: exp(Sum_{k>=1} sigma_5(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301543 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_5(k)).

Original entry on oeis.org

1, 1, 34, 278, 1896, 13074, 92442, 607200, 3866890, 24062327, 146637082, 873517399, 5101981085, 29274370913, 165261721720, 918756928198, 5035250026792, 27229238821726, 145412875008092, 767414597651951, 4004930689994100, 20679955170511834, 105711772783426512

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2331

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), this sequence (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A001160, A301549.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[5, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp((7*Pi)^(6/7) * (Zeta(7)/3)^(1/7) * n^(6/7) / (3*2^(3/7)) - Zeta'(-5)/2) * (Zeta(7)/(3*Pi))^(251/3528) / (2^(251/1176) * 7^(2015/3528) * n^(2015/3528)).

G.f.: exp(Sum_{k>=1} sigma_6(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301544 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_6(k)).

Original entry on oeis.org

1, 1, 66, 796, 7102, 70178, 702813, 6439533, 56938814, 495807251, 4218728690, 34991240657, 284295574638, 2269120791410, 17804772970005, 137455131596032, 1045354069608726, 7839809431539193, 58027706392726849, 424187792875896932, 3064539107659680502

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1769

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), this sequence (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A013954, A301550.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[6, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(8 * 2^(3/8) * Pi * (Zeta(7)/15)^(1/8) * n^(7/8)/7 - Pi*(5/Zeta(7))^(1/8) * n^(1/8) / (504 * 2^(3/8) * 3^(7/8)) + 45*Zeta(7) / (16*Pi^6)) * Zeta(7)^(1/16) / (2^(29/16) * 15^(1/16) * n^(9/16)).

G.f.: exp(Sum_{k>=1} sigma_7(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

Original entry on oeis.org

1, 1, 514, 20198, 414696, 12465714, 373679122, 9181285000, 224372879810, 5583837482767, 132433701077938, 3028947042351535, 68425900639083569, 1518510622688185301, 32936878700790531296, 701684036762210944310, 14726705417058058788172, 304326729686784847885978

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

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

Cf. A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8).

Cf. A013957, A301553.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      sigma[9](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

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

a(n) ~ exp((11*Pi)^(10/11) * (Zeta(11)/3)^(1/11) * n^(10/11) / (2^(3/11) * 5^(10/11)) - Zeta'(-9)/2) * (5*Zeta(11)/(3*Pi))^(131/2904) / (2^(131/968) * 11^(1583/2904) * n^(1583/2904)).

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

A320767 Inverse Euler transform applied once to {1,-1,0,0,0,...}, twice to {1,0,0,0,0,...}, or three times to {1,1,1,1,1,...}.

Original entry on oeis.org

1, 1, -2, 1, -1, 2, -3, 4, -5, 8, -13, 18, -25, 40, -62, 90, -135, 210, -324, 492, -750, 1164, -1809, 2786, -4305, 6710, -10460, 16264, -25350, 39650, -62057, 97108, -152145, 238818, -375165, 589520, -927200, 1459960, -2300346, 3626200, -5720274, 9030450

Offset: 0

Gus Wiseman, Oct 20 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Cf. A000081, A001970, A007562, A007294, A034691, A059966, A061255, A061256, A061257, A065490, A073576, A117209.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
Nest[EulerInvTransform,Array[DiscreteDelta,50,0],2]

cp's OEIS Frontend

A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

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

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

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A301555 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma(k)).

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A320778 Inverse Euler transform of the Euler totient function phi = A000010.

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A301542 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_4(k)).

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A301543 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_5(k)).

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A301544 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_6(k)).

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A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

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

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