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A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

%I A142071 #39 Jan 06 2024 13:03:31
%S A142071 1,0,1,0,1,1,0,1,3,2,0,1,7,12,6,0,1,15,50,60,24,0,1,31,180,390,360,
%T A142071 120,0,1,63,602,2100,3360,2520,720,0,1,127,1932,10206,25200,31920,
%U A142071 20160,5040,0,1,255,6050,46620,166824,317520,332640,181440,40320,0,1,511,18660
%N A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.
%C A142071 Row n gives the coefficients which express the sums of the n-th powers of the integers as a linear combination of binomial coefficients, thus:
%C A142071      Sum_{k=1..r} k^n = A103438(n+r,r) = Sum_{k=0..n} T(n+1,k) * C(r,k),
%C A142071   where, by convention, C(r,k) = 0 whenever r < k. - _Robert B Fowler_, Jan 16 2023
%H A142071 Peter Luschny, <a href="/A142071/b142071.txt">Row 0 to 44, recomputed Jan 19 2019</a>
%F A142071 Row n gives the coefficients of the polynomial defined by p(x, 0) = 1 and for n > 0 p(x, n) = Sum_{k >= 0} k^(n-1)*(x/(1 + x))^k = PolyLog(-n+1, x/(1+x)).
%F A142071 T(n, k) = (k - 1)! * Stirling2(n, k) for k > 0. - _Detlef Meya_, Jan 06 2024
%e A142071 Triangle begins:
%e A142071   1;
%e A142071   0, 1;
%e A142071   0, 1,   1;
%e A142071   0, 1,   3,    2;
%e A142071   0, 1,   7,   12,     6;
%e A142071   0, 1,  15,   50,    60,    24;
%e A142071   0, 1,  31,  180,   390,   360,   120;
%e A142071   0, 1,  63,  602,  2100,  3360,  2520,   720;
%e A142071   0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040;
%e A142071   ...
%p A142071 CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
%p A142071 A142071row := proc(n) 1 - log(1 - x*(exp(z) - 1)):
%p A142071 series(%, z, 12): CL(n!*coeff(%, z, n), x) end:
%p A142071 for n from 0 by 1 to 7 do A142071row(n) od;
%p A142071 # Alternative:
%p A142071 A142071Row := proc(n) if n=0 then [1] else
%p A142071 CL(convert(series(polylog(-n+1, z/(1+z)), z, n*2), polynom), z) fi end:
%p A142071 seq(A142071Row(n), n=0..6); # _Peter Luschny_, Sep 06 2018
%t A142071 T[n_, k_] := If[k==0, Floor[1/(n + 1)], (k - 1)!*StirlingS2[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* _Detlef Meya_, Jan 06 2024 *)
%Y A142071 Column k = 0 is A000007.
%Y A142071 Cf. A028246, A163626, A000629 (row sums).
%Y A142071 Cf. A103438, A007318 (binomial coefficients).
%K A142071 nonn,tabl
%O A142071 0,9
%A A142071 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008
%E A142071 Edited, T(0,0) = 1 prepended and new name by _Peter Luschny_, Sep 06 2018