A002741 -id:A002741 - cp's OEIS frontend

A002743 Sum of logarithmic numbers.

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

A002744 Sum of logarithmic numbers.

Original entry on oeis.org

1, 0, 1, 10, -17, 406, -1437, 20476, -44907, 1068404, -5112483, 230851094, -1942311373, 31916614874, -27260241361, 3826126294680, -37957167335671, 2169009251237640, -25847377785179111, 858747698098918338, -5611513985867158697, 154094365406716365118

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..451
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. A000005, A002746, A318249.

a[n_] := n! * Sum[(-1)^k * DivisorSigma[0, 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)*numdiv(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020

a(n) = Sum_{k=1..n} (-1)^(n-k)*A000005(k)*(k-1)!*binomial(n, k). - Vladeta Jovovic, Feb 09 2003
E.g.f.: -exp(-x) * log(Product_{k>=1} (1 - x^k)^(1/k)). - Ilya Gutkovskiy, Dec 11 2019
a(p) == -2 (mod p) for prime p. The pseudoprimes of this congruence are 4, 6, 20, 42, 1806, ... - Amiram Eldar, May 13 2020

Corrected and extended by Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

A002747 Logarithmic numbers.

Original entry on oeis.org

1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302

Offset: 1

N. J. A. Sloane

sign

abs(a(n)) is also the number of distinct routes starting from a point A and ending at a point B, without traversing any edge more than once, when there are n bi-directional edges connecting A and B. E.g., if there are 3 edges p, q and r from A to B, then the 9 routes starting from A and ending at B are p, q, r, pqr, prq, rpq, rqp, qpr and qrp. - Nikita Kiran, Sep 02 2022

Reducing the sequence modulo the odd integer 2*k + 1 results in a purely periodic sequence with period dividing 4*k + 2, For example, reduced modulo 5 the sequence becomes the purely periodic sequence [1, 3, 4, 2, 0, 4, 2, 1, 3, 0, 1, 3, 4, 2, 0, 4, 2, 1, 3, 0, ...] with period 10. - Peter Bala, Sep 12 2022

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).

Alois P. Heinz, Table of n, a(n) for n = 1..200
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Simon Plouffe, Simple inverter lookup on 1.8134302039235
Simon Plouffe, Smart inverter lookup on 1.8134302039235
Index entries for sequences related to logarithmic numbers

Cf. A000166, A000522, A087208.

a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2013

egf = x/Exp[x]/(1-x^2); a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)
a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 1, 22}] (* Peter Luschny, Dec 18 2017 *)

a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022

E.g.f.: x/exp(x)/(1-x^2). - Vladeta Jovovic, Feb 09 2003
a(n) = n*((n-1)*a(n-2)-(-1)^n). - Matthew Vandermast, Jun 30 2003
From Gerald McGarvey, Jun 06 2004: (Start)
For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.
For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.
For n odd, lim_{n->infinity} a(n)/n! = cosh(1).
For n even, lim_{n->infinity} a(n)/n! = sinh(1).
For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).
For signed values, Sum_{n>=1} a(n)/n!^2 = 0.
For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)
a(n) = (-1)^(n-1)*Sum_{k=0..n} C(n, k)*k!*(1-(-1)^k)/2. - Paul Barry, Sep 14 2004
a(n) = (-1)^(n+1)*n*A087208(n-1). - R. J. Mathar, Jul 24 2015
a(n) = (exp(-1)*Gamma(1+n,-1) - (-1)^n*exp(1)*Gamma(1+n,1))/2 = (A000166(n) - (-1)^n*A000522(n))/2. - Peter Luschny, Dec 18 2017

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

A298374 Expansion of e.g.f. 1/(1 - x)^exp(-x).

Original entry on oeis.org

1, 1, 0, 0, 6, 15, 65, 595, 4004, 32865, 322307, 3316511, 37845214, 471644173, 6319617369, 91114344217, 1404670896264, 23050054222177, 401305630237239, 7387282161642715, 143360257370842146, 2925289119525173741, 62612350725688075941, 1402681525332544374325

Offset: 0

Ilya Gutkovskiy, Jan 18 2018

nonn

Exponential transform of A002741.

			1/(1 - x)^exp(-x) = 1 + x/1! + 6*x^4/4! + 15*x^5/5! + 65*x^6/6! + 595*x^7/7! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..450
N. J. A. Sloane, Transforms
Index entries for sequences related to logarithmic numbers

Cf. A002741, A007116, A009198, A191365.

a:=series(1/(1-x)^exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[1/(1 - x)^Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(-x))) \\ Seiichi Manyama, May 03 2022

a(n) ~ n! * n^(exp(-1)-1) / Gamma(exp(-1)). - Vaclav Kotesovec, May 04 2018

A260322 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.

Original entry on oeis.org

1, -1, 2, 2, -6, 6, 0, 24, -24, 24, 9, -80, 60, -120, 120, 35, 450, 240, 360, -720, 720, 230, -2142, -2310, -840, 2520, -5040, 5040, 1624, 17696, 9744, 21840, -6720, 20160, -40320, 40320, 13209, -112464, 91224, -184464, 15120, -60480, 181440, -362880, 362880

Offset: 1

N. J. A. Sloane, Jul 23 2015

			Triangle begins:
    1;
   -1,     2;
    2,    -6,     6;
    0,    24,   -24,   24;
    9,   -80,    60, -120,  120;
   35,   450,   240,  360, -720,   720;
  230, -2142, -2310, -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 A002741, A002742, A002743, A002744.
Main diagonal gives A000142.
Cf. A260323-A260325.

A260322 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( (-1)^(r-j*n)/(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,",A260322(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True,
   Sum[(-1)^(n-j*k)/(n-j*k)!/j, {j, 1, n/k}]] n!;
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 30 2023 *)

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 *)

A260325 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

Original entry on oeis.org

1, 2, 1, 5, 2, 2, 16, 9, 6, 6, 65, 28, 12, 24, 24, 326, 185, 140, 60, 120, 120, 1957, 846, 750, 120, 360, 720, 720, 13700, 7777, 2562, 5250, 840, 2520, 5040, 5040, 109601, 47384, 47096, 40656, 1680, 6720, 20160, 40320, 40320, 986410, 559953, 378072, 181944, 365904, 15120, 60480, 181440, 362880, 362880

Offset: 1

N. J. A. Sloane, Jul 23 2015

			Triangle begins:
     1;
     2,   1;
     5,   2,   2;
    16,   9,   6,   6;
    65,  28,  12,  24,  24;
   326, 185, 140,  60, 120, 120;
  1957, 846, 750, 120, 360, 720, 720;
  ...

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

Rows, column sums give A000522, A002747, A002750, A002751.
Main diagonal gives A000142.
Cf. A260322, A260323, A260324.

A260325 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n+1)!,j=1..(r+1)/n) ;
        %*r! ;
    end if;
end proc:
for r from 0 to 20 do
    for n from 1 to r+1 do
        printf("%a,",A260325(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True, Sum[1/(n-j*k+1)!, {j, 1, (n+1)/k}]*n!];
Table[T[n, k], {n, 0, 9}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Apr 25 2023 *)

A002742 Logarithmic numbers.

Original entry on oeis.org

2, -6, 24, -80, 450, -2142, 17696, -112464, 1232370, -9761510, 132951192, -1258797696, 20476388114, -225380451870, 4261074439680, -53438049741152, 1151146814425506, -16199301256675974, 391615698778725080, -6109914386833902960

Offset: 1

N. J. A. Sloane

sign

From Peter Bala, Sep 06 2022: (Start)
Conjectures: Let k be a positive integer.
1) for n >= 1, a(n+2*k) - a(n) is divisible by 2*k; if true, then the reduction of the sequence modulo 2*k gives a periodic sequence with period dividing 2*k.
2) for n >= 1, a(n+2*k+1) + a(n) is divisible by 2*k+1; if true, then the reduction of the sequence modulo 2*k+1 gives a periodic sequence with period dividing 4*k + 2. (End)

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).

Alois P. Heinz, Table of n, a(n) for n = 1..200
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Index entries for sequences related to logarithmic numbers

Cf. A002741.

Table[(-1)^(n-1)Sum[Binomial[n+1,2k+1](n-2k)/(k+1)(2k+1)!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Dec 16 2017 *)

makelist((-1)^(n-1)*sum(binomial(n+1,2*k+1)*(n-2*k)/(k+1)*(2*k+1)!,k,0,n),n,0,12); /* Emanuele Munarini, Dec 16 2017 */

first(n) = x='x+O('x^(n+1)); Vec(serlaplace((2*x/(1-x^2)+log(1-x^2))*exp(-x))) \\ Iain Fox, Dec 16 2017

E.g.f.: (2*x/(1-x^2)+log(1-x^2))*exp(-x). - Sean A. Irvine, Aug 11 2014
a(n) = 2*A002747(n) - a(n-1). - R. J. Mathar, Jul 24 2015
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = (-1)^(n-1)*Sum_{k=0..n} binomial(n+1,2*k+1)*((n-2*k)/(k+1))*(2*k+1)!.
a(n+3)+a(n+2)-(n+2)*(n+3)*a(n+1)-(n+2)*(n+3)*a(n) = 2*(-1)^n*(n+3).
(n+3)*a(n+4)+(2*n+7)*a(n+3)-(n+2)*(n+4)^2*a(n+2)-(n+3)*(n+4)*(2*n+5)*a(n+1)-(n+2)*(n+3)*(n+4)*a(n) = 0.
E.g.f.: A(x) = - D(exp(-x)*log(1-x^2)), where D is the derivative with respect to x. (End)
a(n) ~ n! * (exp(-1) - (-1)^n * exp(1)). - Vaclav Kotesovec, Dec 16 2017

More terms from Jeffrey Shallit

More terms from Sean A. Irvine, Aug 11 2014

A002750 Sum of logarithmic numbers.

Original entry on oeis.org

1, 4, 15, 76, 373, 2676, 17539, 152860, 1383561, 14658148, 143131351, 2070738924, 24754959805, 341745565396, 5260157782923, 92358395065276, 1377499388715409, 27622124789948100, 476285499100204831, 10464946811144407948, 222724531608924013701

Offset: 0

N. J. A. Sloane

nonn

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).

Robert Israel, Table of n, a(n) for n = 0..447
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Index entries for sequences related to logarithmic numbers

Cf. A000203.

a:= n-> n!*add(numtheory[sigma](m+1)/(n-m)!, m=0..n):
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 16 2021

a[n_] := n!*Sum[DivisorSigma[1, m+1]/(n-m)!, {m, 0, n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 19 2024, after Alois P. Heinz *)

E.g.f.: F(x)/x*exp(x) where F(x) is o.g.f. for A000203(). a(n) = Sum_{m=1..n+1} A000203(m)*binomial(n, m-1)*(m-1)!. - Vladeta Jovovic, Feb 08 2003

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 08 2003

A007116 Expansion of e.g.f. (1+x)^(exp(x)).

Original entry on oeis.org

1, 1, 2, 6, 18, 75, 295, 1575, 7196, 48993, 230413, 2164767, 8055938, 139431149, 70125991, 14201296057, -77573062280, 2389977322593, -28817693086263, 615493949444827, -10403976760589602, 215611836994976237

Offset: 0

Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A002741, A009198, A191365.

With[{nn=25},CoefficientList[Series[(1+x)^Exp[x],{x,0,nn}], x] Range[ 0,nn]!] (* Harvey P. Dale, Sep 21 2011 *)

a(n):=sum(sum(binomial(n,i)*k^i*stirling1(n-i,k),i,0,n-k),k,1,n); /* Vladimir Kruchinin, Jun 01 2011 */

a(n) = sum(k=1..n, sum(i=0..n-k, binomial(n,i)*k^i*Stirling1(n-i,k))), n>0, a(0)=1. - Vladimir Kruchinin, Jun 01 2011

|a(n)| ~ n!/(Gamma(-exp(-1))*n^(1+exp(-1))). - Vaclav Kotesovec, Jun 27 2013

Definition and terms corrected, and more terms added by Joerg Arndt, Jun 01 2011

cp's OEIS Frontend

A002743 Sum of logarithmic numbers.

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A002744 Sum of logarithmic numbers.

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A002747 Logarithmic numbers.

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A298374 Expansion of e.g.f. 1/(1 - x)^exp(-x).

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A260322 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.

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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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Maple

Mathematica

A260325 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

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