A028342 -id:A028342 - cp's OEIS frontend

A168243 Expansion of e.g.f. Product_{i>=1} (1 + x^i)^(1/i).

1, 1, 1, 5, 11, 59, 439, 2659, 13705, 160649, 2009681, 16966421, 183312931, 2078169235, 34203787591, 657685416179, 8054585463569, 104530824746129, 2595754682459425, 39767021562661669, 758079429084897211

Offset: 0

Vladeta Jovovic, Nov 21 2009

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
Vaclav Kotesovec, Graph: (a(n)/n!) / (n^(log(2) - 1)), 250000 terms
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. A028342.

nmax=20; CoefficientList[Series[Product[(1+x^k)^(1/k),{k,1,nmax}],{x,0,nmax}],x] * Range[0,nmax]! (* Vaclav Kotesovec, May 28 2015 *)
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)

E.g.f.: exp(Sum_{n>=1} A048272(n)*x^n/n).

Conjecture: log(a(n)/n!) ~ (log(2) - 1) * log(n). - Vaclav Kotesovec, Sep 10 2018

A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2n-1))^(-1/(2n-1)).

Original entry on oeis.org

1, 1, 2, 8, 32, 184, 1264, 9568, 79232, 816128, 8769536, 101867776, 1322831872, 18122579968, 268425347072, 4436611211264, 73309336469504, 1303024044310528, 25235367455752192, 497968598916333568, 10431118327503650816, 234674470003955204096, 5359992446798535852032

Offset: 0

Paul D. Hanna, Feb 06 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 32*x^4/4! + 184*x^5/5! + ...
The e.g.f. equals the product:
A(x) = (1-x)^(-1) * (1-x^3)^(-1/3) * (1-x^5)^(-1/5) * (1-x^7)^(-1/7) * (1-x^9)^(-1/9) * (1-x^11)^(-1/11) * ...

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A000005, A001227, A028342, A168243.

m:=40;
f:= func< x | (&*[1/(1 - x^(2*n-1))^(1/(2*n-1)) : n in [1..m+10]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Dec 21 2022

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, 1, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 24 2017

b[n_]:= b[n]= If[n==0, 1, Sum[Sum[If[OddQ[d], 1, 0], {d, Divisors[j]}]* b[n-j], {j, n}]/n];
a[n_]:= b[n]*n!;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

{a(n)=n!*polcoeff(prod(m=1,n,(1-x^(2*m-1)+x*O(x^n))^(-1/(2*m-1))),n)}
for(n=0,31,print1(a(n),", "))

m=40
def f(x): return 1/product( (1 - x^(2*n-1))^(1/(2*n-1)) for n in range(1,m+11) )
def A206303_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A206303_list(m+1) # G. C. Greubel, Dec 21 2022

a(n)/n! is the Euler transform of [1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ...].
E.g.f.: A(x) = B(x) / sqrt(B(x^2)), where B(x) = e.g.f. of A028342.
E.g.f. A(x) satisfies: Product_{n>=0} A(x^(2^n))^(1/2^n) = e.g.f. of A028342.
E.g.f.: exp(Sum_{k>=1} (d(2*k) - d(k))*x^k/k), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Sep 17 2018

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 4, 18, 48, 480, 1440, 20160, 120960, 1451520, 7257600, 239500800, 958003200, 24908083200, 348713164800, 6538371840000, 41845579776000, 2134124568576000, 12804747411456000, 729870602452992000, 9731608032706560000, 204363768686837760000, 2248001455555215360000, 206816133911079813120000

Offset: 1

Ilya Gutkovskiy, Aug 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..450

Column 1 of A338805.

Cf. A000005, A028342, A038048, A318250, A338814.

Table[(n - 1)! DivisorSigma[0, n], {n, 1, 24}]
nmax = 24; Rest[CoefficientList[Series[Sum[Sum[x^(j k)/(j k), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]

a(n) = (n-1)!*numdiv(n); \\ Michel Marcus, Aug 22 2018

E.g.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^(1/k)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A028342.
a(p^k) = (k + 1)*(p^k - 1)!, where p is a prime.

A294462 Expansion of e.g.f. Product_{k>0} (1-k*x^k)^(-1/k).

Original entry on oeis.org

1, 1, 4, 18, 132, 900, 10080, 93240, 1285200, 16526160, 264600000, 3950100000, 81280584000, 1401728328000, 30861115084800, 663835444272000, 16425316331424000, 380082583808928000, 10885891543502976000, 279441709690118976000, 8697410321979899520000

Offset: 0

Seiichi Manyama, Oct 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..424

Column k=1 of A294761.

Cf. A028342, A055225, A294463.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k*x^k)^(1/k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A055225(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 1, 3, 9, 36, 158, 802, 4434, 26978, 176637, 1243528, 9316519, 74065506, 621187700, 5480130494, 50662481722, 489552042241, 4931215686119, 51668848043427, 561981734692781, 6333882472789914, 73850048237680936, 889461218944314524, 11051067390893340510

Offset: 0

Ilya Gutkovskiy, Nov 26 2017

nonn

Exponential transform of A000005.

Seiichi Manyama, Table of n, a(n) for n = 0..553
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; 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
Eric Weisstein's World of Mathematics, Exponential Transform

Cf. A000005, A006171, A028342, A038200, A129921, A160399, A274804, A294363.

a:=series(exp(add(tau(k)*x^k/k!,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] DivisorSigma[0, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A000005(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A000005(k)*a(n-k).

A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 8, 38, 262, 1732, 16144, 153596, 1660796, 19415384, 264084064, 3664187848, 57366995272, 936097392752, 16131362629568, 302946516251408, 6034409270818576, 125044362929875744, 2756094464546395264, 63280996793936902496

Offset: 0

Vaclav Kotesovec, Sep 05 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A088009, A318769, A318696.
Cf. A000262, A305127, A318695.
Cf. A053529, A028342.
Cf. A000010 (phi), A008683 (mu).

nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 29 2022

a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - Ilya Gutkovskiy, Aug 17 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 29 2022
E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - Paul D. Hanna, Jun 24 2023

A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

Original entry on oeis.org

1, 1, 5, 26, 175, 1384, 12933, 135050, 1582901, 20380208, 286577757, 4352682256, 71247772121, 1244923243966, 23166410620637, 456940648889070, 9521696033968393, 208851154175983608, 4812156417656806393, 116112764199821653284, 2928658457243240595901, 77042063713731887400418

Offset: 0

Ilya Gutkovskiy, Sep 16 2021

nonn

Exponential transform of A002746.

Cf. A000005, A002746, A028342, A346546, A346547, A346548.

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002746[n_] := Sum[Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002746[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002746(k) * a(n-k).

A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

Original entry on oeis.org

1, 3, 11, 57, 359, 2793, 25871, 273297, 3268199, 44132313, 659178431, 10710083937, 189256343639, 3636935896233, 75228664345391, 1657133255788977, 38770903634692679, 964609458391250553, 25470259163197390751, 709595190213796188417

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000005, A000629, A002746, A008277, A028342, A308554, A318249, A330352, A330353, A330354, A330445.

nmax = 20; CoefficientList[Series[-Sum[Log[1 - (Exp[x] - 1)^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: Sum_{i>=1} Sum_{j>=1} (exp(x) - 1)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - (exp(x) - 1)^k)^(1/k)).
G.f.: Sum_{k>=1} (k - 1)! * tau(k) * x^k / Product_{j=1..k} (1 - j*x), where tau = A000005.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * tau(k).
a(n) ~ n! * (log(n) + 2*gamma - log(2) - log(log(2))) / (n * (log(2))^n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 14 2019

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 18, 28, 12, 1, 48, 170, 100, 20, 1, 480, 988, 870, 260, 30, 1, 1440, 7896, 7588, 3150, 560, 42, 1, 20160, 60492, 73808, 37408, 9100, 1064, 56, 1, 120960, 555264, 764524, 460656, 140448, 22428, 1848, 72, 1, 1451520, 5819904, 8448120, 5952700, 2162160, 436296, 49140, 3000, 90, 1

Offset: 1

Seiichi Manyama, Nov 10 2020

Also the Bell transform of A318249.

If we use sigma(n,1) in Vladeta Jovovic's formulas in A008298 then one gets the D'Arcais numbers, if we use sigma(n,0) then this sequence arises. # Peter Luschny, Jun 01 2022

			exp(Sum_{n>0} u*d(n)*x^n/n) = 1 + u*x + (2*u+u^2)*x^2/2! + (4*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
      1;
      2,     1;
      4,     6,     1;
     18,    28,    12,     1;
     48,   170,   100,    20,    1;
    480,   988,   870,   260,   30,    1;
   1440,  7896,  7588,  3150,  560,   42,  1;
  20160, 60492, 73808, 37408, 9100, 1064, 56, 1;

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..3 give A318249, A338810, A338811.
Row sums give A028342.
Cf. A000005 (d(n)), A008298, A264428.

# The function BellMatrix is defined in A264428 (with column k = 0).
BellMatrix(n -> n!*NumberTheory:-SumOfDivisors(n+1, 0), 9);
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
(1/n)*x*add(NumberTheory:-SumOfDivisors(n-k, 0)*P(k, x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k = 1..n):
seq(Trow(n), n = 0..10); # Peter Luschny, Jun 01 2022

a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSigma[0, n]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1-x^j+x*O(x^n))^(-u/j)), n), k)}

a(n) = if(n<1, 0, (n-1)!*numdiv(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

E.g.f.: exp(Sum_{n>0} u*d(n)*x^n/n), where d(n) is the number of divisors of n.
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} d(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} d(i_j)/i_j.

cp's OEIS Frontend

A168243 Expansion of e.g.f. Product_{i>=1} (1 + x^i)^(1/i).

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A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(-1/(2*n-1)).

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A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

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A294462 Expansion of e.g.f. Product_{k>0} (1-k*x^k)^(-1/k).

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A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

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A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

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A346545 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).

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A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2n-1))^(-1/(2n-1)).

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).