A061256 -id:A061256 - cp's OEIS frontend

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

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]

A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 8, 8, 5, 1, 1, 1, 16, 21, 21, 7, 1, 1, 1, 32, 56, 84, 39, 11, 1, 1, 1, 64, 153, 331, 206, 92, 15, 1, 1, 1, 128, 428, 1300, 1087, 717, 170, 22, 1, 1, 1, 256, 1221, 5111, 5832, 5512, 1810, 360, 30, 1

Offset: 0

Andrew Howroyd, May 09 2023

T(n,k) is also the number of nonisomorphic (k-1)-tuples of permutations of an n-set that pairwise commute. Isomorphism is up to permutation of the elements of the n-set.

			Array begins:
=======================================================
n/k| 1  2   3    4     5       6        7         8 ...
---+---------------------------------------------------
0  | 1  1   1    1     1       1        1         1 ...
1  | 1  1   1    1     1       1        1         1 ...
2  | 1  2   4    8    16      32       64       128 ...
3  | 1  3   8   21    56     153      428      1221 ...
4  | 1  5  21   84   331    1300     5111     20144 ...
5  | 1  7  39  206  1087    5832    31949    178486 ...
6  | 1 11  92  717  5512   42601   333012   2635637 ...
7  | 1 15 170 1810 19252  208400  2303310  25936170 ...
8  | 1 22 360 5462 81937 1241302 19107225 299002252 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Columns k=1..4 are A000012, A000041, A061256, A226313.

Cf. A160870, A362827, A362903.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
M(n, m=n)={my(v=vector(m), u=vector(n, n, n==1)); for(j=1, #v, v[j]=concat([1], EulerT(u))~; u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(8)); for(n=1, #A~, print(A[n, ])) }

Column k is the Euler transform of column k-1 of A160870.
T(n,k) = A362827(n,k) / n!.
G.f. of column k: exp(Sum_{i>=1} x^i*A160870(i,k)/i).
G.f. of column k > 1: 1/(Product_{i>=1} (1 - x^i)^A160870(i,k-1)).

A006908 Number of nonzero elements in the character table of the symmetric group S_n.

Original entry on oeis.org

1, 4, 8, 21, 39, 92, 170, 331, 593, 1176, 2118, 3699, 6658, 11347, 19760, 32746, 54854, 90245, 149906, 237953, 387937, 608531, 970912, 1510331, 2380015, 3610620, 5634251, 8474110, 12934092, 19440955, 29291690, 43233800, 64825830, 94779612, 139820232

Offset: 1

Simon Plouffe and N. J. A. Sloane

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

J. McKay, personal communication to N. J. A. Sloane, circa 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A006907, A061256.

A006908 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x<>0)); # Eric M. Schmidt, Jul 13 2012, revised Sep 05 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], k_ /; k != 0, 2]; Array[a, 10] (* Jean-François Alcover, Oct 21 2016 *)

More terms from Eric M. Schmidt, Jul 13 2012

A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 1, 4, 8, 24, 42, 118, 208, 524, 961, 2191, 3994, 9020, 16142, 34500, 62814, 130496, 234474, 478334, 855982, 1712012, 3061230, 6003546, 10689178, 20783796, 36789875, 70540531, 124812892, 237022708, 417422168, 786509778, 1381137702, 2583046168, 4526024200, 8402928681

Offset: 0

Ilya Gutkovskiy, Jul 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4150

Cf. A000203, A061256, A180305, A279784, A316962.

with(numtheory): a:=series(mul(1/(1-sigma(k)*x^k),k=1..100),x=0,35): seq(coeff(a,x,n),n=0..34); # Paolo P. Lava, Apr 02 2019

nmax = 34; CoefficientList[Series[Product[1/(1 - DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} sigma(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Jul 28 2018: (Start)
a(n) ~ c * 3^(n/2), where
c = 133.83151651318934683776776253692818185240361972305... if n is even and
c = 131.63961163168586786976253326691345807212512512772... if n is odd.
In closed form, a(n) ~ ((3 + sqrt(3)) * Product_{k>=3} (1/(1 - sigma(k) / 3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3} (1/(1 - (-1)^k * sigma(k) / 3^(k/2)))) * 3^(n/2) / 4. (End)

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

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))).

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)

A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 9, 3, 17, 0, 46, 6, 68, 23, 153, 27, 297, 67, 534, 188, 978, 276, 1932, 620, 3250, 1313, 6033, 2246, 10854, 4361, 18776, 8639, 32831, 14835, 58230, 27635, 98052, 50980, 169522, 88243, 289720, 157179, 486232, 280206, 818006, 478014

Offset: 0

Ilya Gutkovskiy, Oct 20 2019

nonn

Euler transform of A048050.

Convolution of A326830 and A002865 is A318784. - Vaclav Kotesovec, Oct 26 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000203, A001001, A001157, A048050, A061256, A318784, A326831.

with(numtheory):
b:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

nmax = 47; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

G.f.: Product_{k>=1} 1 / (1 - x^k)^A048050(k).
G.f.: exp(Sum_{k>=1} (A001001(k) - A000203(k) - A001157(k) + 1) * x^k / k).
a(n) ~ exp(3^(2/3) * ((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/2 - Pi^2 * (3/((Pi^2 - 6)*Zeta(3)))^(1/3) * n^(1/3)/4 - Pi^4 / (32*(Pi^2 - 6)*Zeta(3)) - 1/8) * A^(3/2)* (2*Pi)^(1/24) / (3^(1/8) * ((Pi^2 - 6)*Zeta(3))^(3/8) * n^(1/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2019

cp's OEIS Frontend

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

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

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

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A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

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A006908 Number of nonzero elements in the character table of the symmetric group S_n.

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A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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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).

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

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