A002750 -id:A002750 - 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 *)

A002748 Sum of logarithmic numbers.

Original entry on oeis.org

1, 2, 3, 26, 13, 1074, -1457, 61802, 7929, 4218722, -6385349, 934344762, -5065189307, 141111736466, 235257551943, 23219206152074, -97011062913167, 11887164842925762, -91890238533000461, 4819930221202545242, -14547510704199530499, 1184314832978574919922

Offset: 0

N. J. A. Sloane

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

Cf. A002750.

A002748 := proc(n) local f1,f2 ; f1 := add(numtheory[sigma](i)*x^(i-1),i=1..n+1) ; f2 := add((-x)^i/i!,i=0..n+1) ; n!*coeftayl(f1*f2,x=0,n) ; end: seq(A002748(n),n=0..25) ; # R. J. Mathar, Oct 22 2007

f1[n_] := Sum[DivisorSigma[1, i]*x^(i-1), {i, 1, n+1}]; f2[n_] := Sum[(-x)^i/i!, {i, 0, n+1}] ; a[n_] := n!*SeriesCoefficient[f1[n]*f2[n], {x, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 17 2014, after R. J. Mathar *)

E.g.f.: F(x)/exp(x)/x where F(x) is o.g.f. for A000203(). - Vladeta Jovovic, Feb 09 2003

More terms from Jeffrey Shallit

More terms from R. J. Mathar, Oct 22 2007

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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