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Search: id:A123125
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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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+0
27
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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; listen)
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OFFSET
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0,9
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COMMENT
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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
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REFERENCES
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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
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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
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EXAMPLE
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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
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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
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CROSSREFS
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See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers.
Adjacent sequences: A123122 A123123 A123124 this_sequence A123126 A123127 A123128
Sequence in context: A099793 A086329 A085852 this_sequence A055105 A058710 A124539
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 30 2006
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EXTENSIONS
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More terms from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 14 2008
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