A061255 -id:A061255 - cp's OEIS frontend

A321264 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^J_n(k), where J_() is the Jordan function.

1, 1, 4, 34, 456, 12388, 677244, 69513187, 13727785600, 5551190294478, 4378921597198116, 6705804947252051188, 21038823519531799964724, 131183284379709847290156854, 1603688086811508900855649976528, 40293997364837932973226463649637881, 2031337795407293560044987268598542021504

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Wikipedia, Jordan's totient function

Cf. A059379, A059380, A061255, A301875, A319647, A321265.

Table[SeriesCoefficient[Product[1/(1 - x^k)^Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[Sum[Sum[d j^n MoebiusMu[d/j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} d*j^n*mu(d/j) ) * x^k/k).

A346770 Expansion of g.f. Product_{k>=1} (1 - x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, -1, -1, -1, 0, 0, 3, 1, 4, 2, 3, -5, 1, -13, -5, -13, -6, -22, 12, -12, 35, 17, 59, 11, 101, -1, 81, -35, 45, -165, 29, -311, -69, -383, -57, -501, 181, -501, 425, -191, 990, -70, 1844, 64, 2305, 183, 2625, -951, 2897, -2701, 1845, -4851, 664, -8824, 670, -12366, 269, -14137, 2884

Offset: 0

Seiichi Manyama, Aug 02 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A061255.

Cf. A000010, A057660, A293116, A299069.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^eulerphi(k)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sigma(k^2, 2)/sigma(k^2)*x^k/k)))

G.f.: exp(-Sum_{k>=1} A057660(k) * x^k/k).

a(0) = 1, a(n) = -(1/n) * Sum_{k=1..n} A057660(k) * a(n-k) for n > 0.

A361721 a(n) = number of isogeny classes of p-divisible groups of abelian varieties of dimension n over an algebraically closed field of characteristic p (for any fixed prime p).

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 20, 31, 47, 70, 103, 151, 218, 313, 446, 629, 883, 1233, 1711, 2362, 3244, 4433, 6034, 8179, 11043, 14852, 19906, 26589, 35400, 46986, 62182, 82057, 107989, 141744, 185583, 242387, 315842, 410627, 532687, 689573, 890837, 1148567, 1478020, 1898430, 2434006, 3115202, 3980232

Offset: 0

Steven Groen, James Rawson, and Robin Visser, Mar 21 2023

nonn

a(n) is the number of p-divisible groups (also called Barsotti-Tate groups) of height 2n which are isomorphic to their own Cartier dual.
The Dieudonné-Manin classification theorem proves that a(n) is the number of symmetric Newton polygons of height 2n and depth n.
S. Harashita proved that log(a(n)) ~ (3/2)*(zeta(3)/zeta(2))^(1/3) * n^(2/3).

			We denote a symmetric Newton polygon of height 2n and depth n as a sequence of nonnegative integer coordinates: (0,0)--(x1,y1)--(x2,y2)--...--(xk,yk)--(2n,n) such that the slope of the line through (xi, yi), (x_{i+1}, y_{i+1}) is strictly less than the slope of the line through (x_{i+1}, y_{i+1}), (x_{i+2}, y_{i+2}), and such that, for any 0 < x < 2n, the slope at x plus the slope at 2n-x equals 1.
For n = 2, the a(2) = 3 possible symmetric Newton polygons of length 4 and depth 2 are:
 (0,0)--(4,2)
 (0,0)--(2,0)--(4,2)
 (0,0)--(1,0)--(3,1)--(4,2)
For n = 3, the a(3) = 5 possible symmetric Newton polygons of length 6 and depth 3 are:
 (0,0)--(6,3)
 (0,0)--(3,0)--(6,3)
 (0,0)--(3,1)--(6,3)
 (0,0)--(2,0)--(4,1)--(6,3)
 (0,0)--(1,0)--(5,2)--(6,3)

Robin Visser, Table of n, a(n) for n = 0..200
Y. W. Ding and Y. Ouyang, A simple proof of Dieudonné-Manin classification theorem, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 8, 1553-1558.
S. Harashita, Asymptotic formula of the number of Newton polygons, Math. Z. 297 (2021), no. 1-2, 113-132.
M. Rapoport, On the Newton stratification, Astérisque No. 290 (2003), Séminaire Bourbaki, Exp. No. 903, viii, 207-224.

Cf. A061255.

# Use generating function to return a(n)
def a(n):
    f = product([(1 - x^k)^(-euler_phi(k)) for k in range(1,n+1)])
    gf = sqrt((1+x)*f)/(1-x)
    return gf.taylor(x,0,n).coefficients()[n][0]

G.f.: sqrt((1+x)*f(x))/(1-x) where f(x) = Product_{k>=1} (1 - x^k)^(-phi(k)).

a(n) ~ 2*K^(1/2) / (sqrt(6*Pi) * C^(7/36) * (2*n)^(11/36)) * exp((3/4)*C^(1/3) * (2n)^(2/3) + (1/2)*(Sum_a g_a(C^(1/3) * (2n)^(-1/3)))), where C = 2*zeta(3)/zeta(2), K = exp(-2*zeta'(-1) - log(2*Pi)/6), g_a(x) is the residue of Gamma(s)*zeta(s+1)*zeta(s-1)/(zeta(s)*x^s) at s=a, and where Sum_a runs through all nontrivial zeros a of zeta(s) [Harashita].

A381680 Euler transform of A115224.

Original entry on oeis.org

1, 1, 29, 263, 1565, 11217, 74412, 482638, 2987123, 18066149, 107415185, 623612637, 3552605428, 19882256022, 109518424910, 594290145192, 3179607733480, 16790129919934, 87573088547032, 451477766533886, 2302069862201553, 11616226357007259, 58036597014533469

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A061255, A381679.

Cf. A115224, A084220.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[6, k^2]/DivisorSigma[3, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 6)/sigma(k^2, 3)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A115224(k).
G.f.: exp( Sum_{k>=1} sigma_6(k^2)/sigma_3(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_6(k^2)/sigma_3(k^2) * a(n-k).
log(a(n)) ~ 7 * 5^(2/7) * zeta(7)^(1/7) * n^(6/7) / (2^(2/7) * 3^(3/7) * Pi^(4/7)). - Vaclav Kotesovec, Mar 04 2025

A307705 Expansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 8, 5, 16, 15, 34, 30, 75, 66, 144, 150, 285, 292, 566, 585, 1062, 1170, 1988, 2205, 3729, 4159, 6755, 7785, 12214, 14147, 21957, 25560, 38709, 45839, 67884, 80747, 118332, 141244, 203614, 245330, 348396, 420971, 592439, 717659, 998248, 1215439, 1672544, 2040210, 2786687

Offset: 0

Ilya Gutkovskiy, Apr 22 2019

nonn

Euler transform of A051953.

Cf. A000010, A001157, A051953, A053650, A057660, A061255, A065764, A065827.

nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(k - EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[(DivisorSigma[2, k] - DivisorSigma[2, k^2]/DivisorSigma[1, k^2]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 - EulerPhi[d^2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} (sigma_2(k) - sigma_2(k^2)/sigma_1(k^2)) * x^k/k).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} cototient(d^2) ) * x^k/k).
a(n) ~ exp(3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) + 1/4) * ((Pi^2 - 6)*Zeta(3))^(1/4) / (A^3 * 2^(1/12) * 3^(1/2) * Pi^(5/6) * n^(3/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 06 2019

A320783 Inverse Euler transform of (-1)^(n - 1).

Original entry on oeis.org

1, 1, -2, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986, -505294125, 981706806

Offset: 0

Gus Wiseman, Oct 22 2018

sign

After a(1) and a(2), same as A038063.

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.

OEIS Wiki, 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, A320778, A320779, A320780, A320781, A320782.

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[(-1)^(#-1)&,30]]

A320785 Inverse Euler transform of the number of factorizations function A001055.

Original entry on oeis.org

1, 1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 1, -1, 0, 0, 1, 1, -3, 3, -3, 0, 4, -6, 6, -5, 5, -1, -7, 13, -16, 15, -8, -3, 12, -25, 41, -40, 21, 10, -51, 83, -93, 81, -38, -44, 148, -234, 258, -190, 35, 184, -429, 616, -660, 480, -18, -640, 1289, -1714, 1693, -1039, -268

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.

OEIS Wiki, 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, A320778, A320779, A320780, A320781, A320782.

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]}]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
EulerInvTransform[Table[Length[facs[n]],{n,100}]]

A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -2, 2, -2, 3, -5, 6, -7, 11, -16, 20, -27, 39, -55, 75, -102, 145, -207, 286, -397, 565, -802, 1123, -1581, 2248, -3193, 4517, -6399, 9112, -12984, 18457, -26270, 37502, -53553, 76416, -109146, 156135, -223446, 319764, -457884, 656288, -941081

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.

OEIS Wiki, 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, A320778, A320779, A320780, A320781, A320782.

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[PadRight[{1,0,1},50]]

cp's OEIS Frontend

A321264 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^J_n(k), where J_() is the Jordan function.

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A346770 Expansion of g.f. Product_{k>=1} (1 - x^k)^phi(k), where phi() is the Euler totient function (A000010).

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PARI

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A361721 a(n) = number of isogeny classes of p-divisible groups of abelian varieties of dimension n over an algebraically closed field of characteristic p (for any fixed prime p).

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A381680 Euler transform of A115224.

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A307705 Expansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

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Mathematica

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A320783 Inverse Euler transform of (-1)^(n - 1).

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A320785 Inverse Euler transform of the number of factorizations function A001055.

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A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

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