A260324 -id:A260324 - cp's OEIS frontend

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

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 *)

A002749 Sum of logarithmic numbers.

Original entry on oeis.org

1, 1, 1, 11, -7, 389, -1031, 19039, -24431, 1023497, -4044079, 225738611, -1711460279, 29974303501, 4656373513, 3798866053319, -34131041040991, 2131052083901969, -23678368533941471, 832900320313739227, -4752766287768240359, 148482851420849206421

Offset: 0

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).

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Index entries for sequences related to logarithmic numbers

A002749 := proc(r)
    add(A260324(n,r),n=1..r+1) ;
end proc:
seq(A002749(r),r=0..25) ; # R. J. Mathar, Jul 24 2015

m = 22;
F[x_] = Sum[DivisorSigma[0, n] x^n , {n, 1, m}];
CoefficientList[F[x]/(x E^x) + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Mar 30 2020 *)

E.g.f.: F(x)/exp(x)/x where F(x) is o.g.f. for A000005(). - Vladeta Jovovic, Feb 09 2003

Corrected and extended by Jeffrey Shallit

cp's OEIS Frontend

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

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