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A008518
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Triangle of Eulerian numbers with rows multiplied by 1 + x.
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16
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1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1, 1, 1014
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OFFSET
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0,5
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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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FORMULA
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E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 2, 1;
1, 5, 5, 1;
1, 12, 22, 12, 1;
1, 27, 92, 92, 27, 1;
1, 58, 359, 604, 359, 58, 1;
1, 121, 1311, 3607, 3607, 1311, 121, 1;
...
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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_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
T[n_, k_] := A[n, k] + A[n, k+1];
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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