A260324 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

1, 0, 1, 1, -2, 2, 2, 9, -6, 6, 9, -28, 12, -24, 24, 44, 185, 100, 60, -120, 120, 265, -846, -690, -120, 360, -720, 720, 1854, 7777, 2478, 5250, -840, 2520, -5040, 5040, 14833, -47384, 33656, -40656, 1680, -6720, 20160, -40320, 40320, 133496, 559953, -347832, 181944, 359856, 15120, -60480, 181440, -362880, 362880

Offset

1,5

Links

Table of n, a(n) for n=1..55.

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

Example

Triangle begins:
1,
0,1,
1,-2,2,
2,9,-6,6,
9,-28,12,-24,24,
44,185,100,60,-120,120,
265,-846,-690,-120,360,-720,720,
...

Maple

A260324 := proc(n, r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( (-1)^(r-j*n+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, ", A260324(n, r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

Mathematica

T[n_, k_] := If[k == 0, 1, If[n > k + 1, 0, k! Sum[(-x)^(k - j n + 1)/(k - j n + 1)!, {j, 1, (k + 1)/n}]]];
Table[T[n, k] /. x -> 1, {k, 0, 9}, {n, 1, k + 1}] // Flatten (* Jean-François Alcover, Mar 30 2020 *)

Crossrefs

Rows, column sums give A000166, A002747, A002748, A002749.

Cf. A260322, A260323, A260325.

Sequence in context: A260662 A228044 A171529 * A157649 A155695 A195706

Adjacent sequences: A260321 A260322 A260323 * A260325 A260326 A260327

Keyword

sign,tabl

Author

N. J. A. Sloane, Jul 23 2015

Status

approved