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