A330354
Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).
Original entry on oeis.org
1, 2, 1, 21, -122, 1752, -21730, 309166, -4521032, 70344768, -1173530712, 21642745704, -448130571696, 10352684535840, -260101132095888, 6921279885508848, -191813249398678272, 5502934340821289088, -163695952380982280832, 5078687529186002247552
Offset: 1
Cf.
A000041,
A000203,
A002743,
A008275,
A038048,
A089064,
A306042,
A330351,
A330352,
A330353,
A330494.
-
nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
A002745
Sum of logarithmic numbers.
Original entry on oeis.org
1, 5, 20, 96, 469, 3145, 20684, 173544, 1557105, 16215253, 159346604, 2230085528, 26985045333, 368730610729, 5628888393652, 97987283458928, 1475486672174337, 29097611462122437, 505383110562327268, 10970329921706735216
Offset: 1
- J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
- Jeffrey Shallit, personal communication.
- 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
-
Table[Sum[Binomial[n,k] * DivisorSigma[1,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 16 2019 *)
-
a(n) = sum(k=1, n, sigma(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020
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 *)
A346548
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(-x).
Original entry on oeis.org
1, 1, 2, 6, 42, 175, 2015, 10843, 157388, 1240377, 20118077, 172029231, 4052166250, 36360150385, 952965601471, 11194257455977, 316421367496344, 3722989943371217, 134504815853036649, 1641201826969536379, 67298415781492985366, 935342610632498431241, 40176825083871581430723
Offset: 0
-
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Exp[-x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002743[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002743[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
Showing 1-4 of 4 results.
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