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A000800
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Sum of upward diagonals of Eulerian triangle.
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5
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1, 1, 1, 2, 5, 13, 38, 125, 449, 1742, 7269, 32433, 153850, 772397, 4088773, 22746858, 132601933, 807880821, 5132235182, 33925263901, 232905588441, 1657807491222, 12215424018837, 93042845392105, 731622663432978, 5931915237693517
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OFFSET
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0,4
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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G.f.: 1/(1-x/(1-x^2/(1-2x/(1-2x^2/(1-3x/(1-3x^2/(1-... (continued fraction). [Paul Barry, Mar 24 2010]
a(n) = Sum_{k} A173018(n-k, k) - Michael Somos, Mar 17 2011
G.f.: 1/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x^2*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 14 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - x*(x+1)*k - x^3*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 14 2013
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EXAMPLE
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1 = 1, 1 = 1, 1 = 1 + 0, 2 = 1 + 1, 5 = 1 + 4 + 0, etc.
1 + x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 38*x^6 + 125*x^7 + 449*x^8 + 1742*x^9 + ...
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MATHEMATICA
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t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, k_] := t[n, k] = (n-k)*t[n-1, k-1] + (k+1)*t[n-1, k]; a[n_] := Sum[t[n-k, k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 14 2011, after Michael Somos *)
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CROSSREFS
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Cf. A173018.
Sequence in context: A148303 A148304 A149859 * A149860 A006823 A151446
Adjacent sequences: A000797 A000798 A000799 * A000801 A000802 A000803
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Tony Harkin [ harkin(AT)mit.edu, tharkin(AT)vortex.weather.brockport.edu ]
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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