A301545 -id:A301545 - cp's OEIS frontend

A061256 Euler transform of sigma(n), cf. A000203.

1, 1, 4, 8, 21, 39, 92, 170, 360, 667, 1316, 2393, 4541, 8100, 14824, 26071, 46422, 80314, 139978, 238641, 408201, 686799, 1156062, 1920992, 3189144, 5238848, 8589850, 13963467, 22641585, 36447544, 58507590, 93334008, 148449417, 234829969, 370345918

Offset: 0

Vladeta Jovovic, Apr 21 2001

This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by Franklin T. Adams-Watters, Jan 16 2006, and proved by J. R. Britnell in 2012.
According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

			1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
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.
J. R. Britnell, A formal identity involving commuting triples of permutations, arXiv:1203.5079 [math.CO], 2012.
J. R. Britnell, A formal identity involving commuting triples of permutations, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
J. R. Britnell, A formal identity involving commuting triples of permutations, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
E. Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
Secret Blogging Seminar, A peculiar numerical coincidence.
N. J. A. Sloane, Transforms
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 18 2012 *)
nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

a(n) = A072169(n) / n!.
G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [Paul D. Hanna, Mar 28 2009]
G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [Vladeta Jovovic, Mar 28 2009]
G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [Joerg Arndt, Apr 12 2013]
a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018

Entry revised by N. J. A. Sloane, Jun 13 2012

A275585 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

Original entry on oeis.org

1, 1, 6, 16, 52, 128, 373, 913, 2399, 5796, 14298, 33655, 79756, 183078, 419846, 942807, 2106176, 4633208, 10127557, 21870997, 46912648, 99639685, 210206722, 439777198, 914157490, 1886428608, 3869204040, 7884691072, 15976273573, 32182538964, 64484592372, 128518359868, 254868985099, 502950483815, 987904826874, 1931596634076

Offset: 0

Ilya Gutkovskiy, Dec 25 2016

nonn

Euler transform of the sum of squares of divisors (A001157).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to sums of divisors

Cf. A001157, A027847, A288414.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), this sequence (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[2](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nmax = 35; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[2, k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 08 2017
a(n) ~ exp(4*Pi * Zeta(3)^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi * 5^(1/4) * n^(1/4) / (8 * 3^(7/4) * Zeta(3)^(1/4)) + Zeta(3) / (8*Pi^2)) * Zeta(3)^(1/8) / (2^(3/2) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

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

Original entry on oeis.org

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))).

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

A301551 Expansion of Product_{k>=1} (1 + x^k)^(sigma_7(k)).

Original entry on oeis.org

1, 1, 129, 2317, 26957, 385147, 5514889, 70250881, 866874825, 10634404922, 126906497939, 1470673175003, 16705788322140, 186487470519166, 2044203433733016, 22025647881901542, 233686866722213324, 2443978994099801452, 25211475391206919299, 256716054713570158748

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

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

Cf. A013955, A107742, A301545.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[7, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(9 * Pi^(8/9) * (17*Zeta(9))^(1/9) * n^(8/9) / 2^(29/9)) * (17*Zeta(9)/Pi)^(1/18) / (3 * 2^(883/1440) * n^(5/9)).

G.f.: exp(Sum_{k>=1} sigma_8(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 8, 11, 1, 1, 10, 16, 21, 17, 1, 1, 18, 38, 52, 39, 34, 1, 1, 34, 100, 156, 128, 92, 52, 1, 1, 66, 278, 526, 534, 373, 170, 94, 1, 1, 130, 796, 1896, 2546, 2014, 913, 360, 145, 1, 1, 258, 2318, 7102, 13074, 12953, 6796, 2399, 667, 244

Offset: 0

Ilya Gutkovskiy, Nov 20 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
  11,  21,   52,  156,   526,   1896,  ...
  17,  39,  128,  534,  2546,  13074,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..9 give A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547.
Main diagonal gives A319647.
Cf. A321877.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^j)), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

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

G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^j))).

cp's OEIS Frontend

A061256 Euler transform of sigma(n), cf. A000203.

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A275585 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

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

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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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A301551 Expansion of Product_{k>=1} (1 + x^k)^(sigma_7(k)).

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