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A059297
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Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.
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8
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1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
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LINKS
| P. Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983)
E. W. Weisstein, MathWorld: Abel Polynomial
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FORMULA
| E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 18 2003
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011
The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011
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EXAMPLE
| Triangle begins 1; 0, 1; 0, 2, 1; 0, 3, 6, 1; 0, 4, 24, 12, 1; ...
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....
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CROSSREFS
| There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248. A061356. A201017.
Sequence in context: A202178 A035543 A105546 * A077874 A153007 A090683
Adjacent sequences: A059294 A059295 A059296 * A059298 A059299 A059300
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 25 2001
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