%I A123125
%S A123125 1,0,1,0,1,1,0,1,4,1,0,1,11,11,1,0,1,26,66,26,1,0,1,57,302,302,57,1,0,
               1,
%T A123125 120,1191,2416,1191,120,1,0,1,247,4293,15619,15619,4293,247,1,0,1,502,
%U A123125 14608,88234,156190,88234,14608,502,1,0,1,1013,47840,455192,1310354
%N A123125 Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows.
%C A123125 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,0,2,0,3,0,4,0,5,
               0,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,...] where DELTA is the operator 
               defined in A084938.
%C A123125 Row sums are the factorials. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Aug 14 2008
%C A123125 If the initial zero column is deleted, the result is like Pascal's triangle. 
               - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Aug 14 2008
%C A123125 This result gives an alternative method of calculating the Eulerian numbers 
               by an Umbral Calculus expansion from Comtet. Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Nov 21 2009
%C A123125 This function seems to be equivalent to the PolyLog expansion. [From 
               Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009]
%D A123125 Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series 
               Analysis, MaGraw-Hill, New York, 1976, page 91 - from Roger L. Bagula 
               and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008
%D A123125 L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245 [From 
               Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009]
%F A123125 Sum{k,0<=k<=n}T(n,k)=n!=A000142(n) . Sum{k,0<=k<=n}2^k*T(n,k)=A000629(n) 
               . Sum{k,0<=k<=n}3^k*T(n,k)=abs(A009362(n+1)) . Sum{k,0<=k<=n}2^(n-k)*T(n,
               k)=A000670(n).
%F A123125 Sum_{k, 0<=k<=n}T(n,k)*3^(n-k)=A122704(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 07 2007
%F A123125 G.f.: f(x,n)=(1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}] - Roger L. 
               Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008
%e A123125 Triangle begins:
%e A123125 {1},
%e A123125 {0, 1},
%e A123125 {0, 1, 1},
%e A123125 {0, 1, 4, 1},
%e A123125 {0, 1, 11, 11, 1},
%e A123125 {0, 1, 26, 66, 26, 1},
%e A123125 {0, 1, 57, 302, 302, 57, 1},
%e A123125 {0, 1, 120, 1191, 2416, 1191, 120, 1},
%e A123125 {0, 1, 247, 4293, 15619, 15619, 4293, 247, 1},
%e A123125 {0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1},
%e A123125 {0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1}
%e A123125 ...
%t A123125 f[x_, n_] := f[x, n] = (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}]; 
               Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[f[x\
               , n]]], x], {n, 0, 10}]; Flatten[a] - Roger L. Bagula and Gary W. 
               Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008
%t A123125 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 
               2009: (Start)
%t A123125 Clear[p, g, m, a];
%t A123125 p[t_] = (1 - x)/(1 - x*Exp[t*(1 - x)])'
%t A123125 a = Table[ CoefficientList[ FullSimplify[ ExpandAll[ n!*SeriesCoefficient[ 
               Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A123125 Flatten[a] (End)
%Y A123125 See A008292 (subtriangle for k>=1 and n>=1), which is the main entry 
               for these numbers.
%Y A123125 Sequence in context: A099793 A086329 A085852 this_sequence A055105 A058710 
               A124539
%Y A123125 Adjacent sequences: A123122 A123123 A123124 this_sequence A123126 A123127 
               A123128
%K A123125 nonn,tabl,new
%O A123125 0,9
%A A123125 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 30 2006
%E A123125 More terms from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Aug 14 2008