A002745 -id:A002745 - cp's OEIS frontend

A346547 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).

1, 1, 6, 36, 282, 2575, 28075, 340809, 4657996, 69874305, 1145441713, 20279904337, 386803154474, 7874727448757, 170678885319787, 3919163707551187, 95029714996046680, 2424604353738271201, 64940619086990938317, 1820746123923294245293, 53328181409328560026038

Offset: 0

Ilya Gutkovskiy, Sep 16 2021

nonn

Exponential transform of A002745.

Cf. A000203, A002745, A053529, A346545, A346546, A346548.

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002745[n_] := Sum[Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002745[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002745(k) * a(n-k).

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

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

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

A002743 Sum of logarithmic numbers.

Original entry on oeis.org

1, 1, 2, 24, -11, 1085, -2542, 64344, -56415, 4275137, -10660486, 945005248, -6010194555, 147121931021, 88135620922, 23131070531152, -120142133444319, 12007306976370081, -103897545509370542, 4923827766711915784, -19471338470911446283, 1203786171449486366205

Offset: 1

N. J. A. Sloane

sign

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..449
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
J. M. Gandhi, Logarithmic Numbers and the Functions d(n) and sigma(n), The American Mathematical Monthly, Vol. 73, No. 9 (1966), pp. 959-964, alternative link.
Index entries for sequences related to logarithmic numbers

Cf. A000203, A002745.

a[n_] := n! * Sum[(-1)^k * DivisorSigma[1, n - k]/k!/(n - k), {k, 0, n - 1}]; Array[a, 22] (* Amiram Eldar, May 13 2020 *)

a(n) = sum(k=1, n, (-1)^(n-k)*sigma(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020

a(n) = Sum_{k=1..n} (-1)^(n-k)*A000203(k)*(k-1)!*binomial(n, k). - Vladeta Jovovic, Feb 09 2003
E.g.f.: exp(-x) * Sum_{k>=1} x^k / (k*(1 - x^k)). - Ilya Gutkovskiy, Dec 11 2019
a(p) == -1 (mod p) for prime p. The pseudoprimes of this congruence are 6, 42, 1806, ... - Amiram Eldar, May 13 2020

More terms from Jeffrey Shallit

A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

Original entry on oeis.org

1, 3, 2, 8, 6, 6, 24, 24, 24, 24, 89, 80, 60, 120, 120, 415, 450, 480, 360, 720, 720, 2372, 2142, 2730, 840, 2520, 5040, 5040, 16072, 17696, 10416, 21840, 6720, 20160, 40320, 40320, 125673, 112464, 151704, 184464, 15120, 60480, 181440, 362880, 362880

Offset: 1

N. J. A. Sloane, Jul 23 2015

			Triangle begins:
1,
3,2,
8,6,6,
24,24,24,24,
89,80,60,120,120,
415,450,480,360,720,720,
2372,2142,2730,840,2520,5040,5040,
...

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Rows, column sums give A002104, A002742, A002745, A002746.

Cf. A260322-A260325.

A260323 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n)!/j,j=1..(r)/n) ;
        %*r! ;
    end if;
end proc:
for r from 1 to 20 do
    for n from 1 to r do
        printf("%a,",A260323(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

T[n_, k_] := If[n == 0, 1, If[k > n+1, 0, Sum[1/(n - j*k)!/j, {j, 1, n/k}]]]*n!;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar *)

A356589 a(n) = n! * Sum_{k=1..n} sigma_k(k)/(k * (n-k)!).

Original entry on oeis.org

1, 7, 74, 1896, 83829, 6169915, 634444586, 89796130088, 16407420884385, 3792452363345383, 1076168167972120354, 368657061467873013440, 149787334364400115372677, 71262783791831946810277899, 39228224120114488162020163762

Offset: 1

Seiichi Manyama, Aug 14 2022

nonn

Cf. A002745, A002746, A356437, A356590.

a[n_] := n! * Sum[DivisorSigma[k, k]/(k*(n - k)!), {k, 1, n}]; Array[a, 15] (* Amiram Eldar, Aug 14 2022 *)

a(n) = n!*sum(k=1, n, sigma(k, k)/(k*(n-k)!));

my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-(k*x)^k)/k)))

E.g.f.: -exp(x) * Sum_{k>0} log(1 - (k*x)^k)/k.

a(n) ~ n! * n^(n-1). - Vaclav Kotesovec, Aug 17 2022

A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

Original entry on oeis.org

1, 7, 38, 240, 1509, 12115, 96326, 929432, 9421089, 108909943, 1249105054, 17862483320, 241674418101, 3676733397363, 59149265744302, 1058605924855568, 18041587282787489, 363409114370324295, 6970858463185187062, 153017341796727034336, 3360005220780469981157

Offset: 1

Seiichi Manyama, Aug 15 2022

nonn

The average value of a(n) is zeta(3) * exp(1) * n * n!. - Vaclav Kotesovec, Aug 17 2022

Cf. A002745, A002746, A356589.

Cf. A356298.

Table[n! * Sum[DivisorSigma[2, k]/(k * (n-k)!), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 17 2022 *)

a(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, k*log(1-x^k))))

E.g.f.: exp(x) * Sum_{k>0} x^k/(k * (1 - x^k)^2).

E.g.f.: -exp(x) * Sum_{k>0} k * log(1 - x^k).

cp's OEIS Frontend

A346547 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).

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A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

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A002743 Sum of logarithmic numbers.

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A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

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A356589 a(n) = n! * Sum_{k=1..n} sigma_k(k)/(k * (n-k)!).

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A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

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