A002746
Sum of logarithmic numbers.
Original entry on oeis.org
1, 4, 13, 50, 203, 1154, 6627, 49356, 403293, 3858376, 33929377, 460614670, 5168544119, 64518640406, 946910125319, 16124114481720, 221243980745433, 4261440137319852, 68524390012831189, 1477309421907315082
Offset: 1
- 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..450
- 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
-
Table[Sum[Binomial[n,k] * DivisorSigma[0,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 16 2019 *)
-
a(n) = sum(k=1, n, numdiv(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020
A330352
Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.
Original entry on oeis.org
1, 1, 0, 10, -68, 818, -9782, 130730, -1835752, 27408672, -438578616, 7697802264, -150743052528, 3293454634416, -78787556904864, 2014008113598432, -54001416897306240, 1504891127666322048, -43527807706621236480, 1311515508480252542208
Offset: 1
-
nmax = 20; CoefficientList[Series[-Sum[Log[1 - Log[1 + x]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]
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
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]
-
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 *)
A346546
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(-x)/k).
Original entry on oeis.org
1, 1, 1, 2, 15, 44, 485, 1854, 25781, 170288, 2477485, 12571140, 435748665, 2049818198, 64651106637, 628176476186, 18837010964105, 93248340364152, 6695745240354169, 33794005826851192, 2549048418922818525, 20209158430316698922, 1138228671555859916609
Offset: 0
-
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[-x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002744[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002744[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
Showing 1-4 of 4 results.
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