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A123125
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Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows.
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39
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| The beginning of this sequence does not quite agree with the usual version, which is A173018. - N. J. A. Sloane, Nov 21 2010
Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos Mar 17 2011
A008292 (subtriangle for k>=1 and n>=1 is the main entry for these numbers.
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]
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REFERENCES
| Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, Arxiv preprint arXiv:1105.3043, 2011
L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245 [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
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FORMULA
| 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
Sum _{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A000142(n), A000629(n), A123227(n), A201355(n), A201368(n) for x = 0, 1, 2, 3, 4, 5 respectively. - DELEHAM Philippe, Dec 01 2011
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EXAMPLE
| 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}
...
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MATHEMATICA
| 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)
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CROSSREFS
| See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers. Another version has the zeros at the ends of the rows, as in Concrete Mathematics: see A173018.
Sequence in context: A099793 A086329 A085852 * A173018 A055105 A200545
Adjacent sequences: A123122 A123123 A123124 * A123126 A123127 A123128
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 30 2006
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EXTENSIONS
| More terms from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008
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