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A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].

%I A174859 #4 Jul 22 2025 07:55:23
%S A174859 1,0,1,0,1,-1,0,1,0,-5,0,1,3,-16,15,0,1,10,-40,25,56,0,1,25,-81,-30,
%T A174859 370,-455,0,1,56,-119,-469,1841,-1960,-237,0,1,119,-22,-2527,7448,
%U A174859 -5768,-7420,16947,0,1,246,766,-10359,24627,-2289,-76692,126504,-64220,0,1
%N A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].
%C A174859 Row sums are:
%C A174859 {1, 1, 0, -4, 3, 52, -170, -887, 8778, -1416, -415734,...}.
%D A174859 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 77.
%F A174859 p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];
%F A174859 t(n,m)=coefficients(p(x,n))
%e A174859 {1},
%e A174859 {0, 1},
%e A174859 {0, 1, -1},
%e A174859 {0, 1, 0, -5},
%e A174859 {0, 1, 3, -16, 15},
%e A174859 {0, 1, 10, -40, 25, 56},
%e A174859 {0, 1, 25, -81, -30, 370, -455},
%e A174859 {0, 1, 56, -119, -469, 1841, -1960, -237},
%e A174859 {0, 1, 119, -22, -2527, 7448, -5768, -7420, 16947},
%e A174859 {0, 1, 246, 766, -10359, 24627, -2289, -76692, 126504, -64220},
%e A174859 {0, 1, 501, 4265, -36320, 60215, 119760, -570627, 784245, -248280, -529494}
%t A174859 Clear[p, x, n];
%t A174859 p[x_, n_] = Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];
%t A174859 Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t A174859 Flatten[%]
%Y A174859 Cf. A008299
%K A174859 sign,tabl,uned
%O A174859 0,10
%A A174859 _Roger L. Bagula_, Mar 31 2010