0,9

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.

Row sums are the factorials. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008

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

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

This function seems to be equivalent to the PolyLog expansion. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009]

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

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009]

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).

Sum_{k, 0<=k<=n}T(n,k)*3^(n-k)=A122704(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 07 2007

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

Triangle begins:

{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, 120, 1191, 2416, 1191, 120, 1},

{0, 1, 247, 4293, 15619, 15619, 4293, 247, 1},

{0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1},

{0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1}

...

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

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009: (Start)

Clear[p, g, m, a];

p[t_] = (1 - x)/(1 - x*Exp[t*(1 - x)])'

a = Table[ CoefficientList[ FullSimplify[ ExpandAll[ n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];

Flatten[a] (End)

See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers.

Sequence in context: A099793 A086329 A085852 this_sequence A055105 A058710 A124539

Adjacent sequences: A123122 A123123 A123124 this_sequence A123126 A123127 A123128

nonn,tabl,new

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 30 2006

More terms from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008