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A046739
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Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
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5
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0, 1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 161, 51, 1, 1, 113, 813, 813, 113, 1, 1, 239, 3361, 7631, 3361, 239, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 1, 2025, 141549, 1704693, 5494017
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009: (Start)
T(n,k) is the number of derangements of [n] having k excedances. Example: T(4,2)=7 because we have 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 3*4*21, 4*3*21, each with two excedances (marked). An excedance of a permutation p is a position i such that p(i)>i.
Sum(k*T(n,k),k>=1)=A000274(n+1).
(End)
The triangle 1;1,1;1,7,1;... has general term T(n,k)=sum{j=0..n+2, (-1)^(n-j)*C(n+2,j)*A123125(j,k+2)} and bivariate g.f. ((1-y)*(y*exp(2*x*y)+exp(x*(y+1))(y^2-4*y+1)+y*exp(2*x)))/(exp(x*y)-y*exp(x))^3. [Paul Barry, May 10 2011]
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REFERENCES
| D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]
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FORMULA
| a(n+1, r)=r*a(n, r)+(n+1-r)a(n, r-1)+n*a(n-1, r-1).
exp(-t)/(1 - exp((x-1)t)/(x-1)) = 1 + x*t^2/2! + (x+x^2)*t^3/3! + (x+7x^2+x^3)*t^4/4! + (x+21x^2+21x^3+x^4)*t^5/5! + ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 11 2004
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EXAMPLE
| 0; 1; 1 1; 1 7 1; 1 21 21 1; 1 51 161 51 1; ...
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MAPLE
| G := (1-t)*exp(-t*z)/(1-t*exp((1-t)*z)): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]
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MATHEMATICA
| max = 12; f[t_, z_] := (1-t)*(Exp[-t*z]/(1 - t*Exp[(1-t)*z])); se = Series[f[t, z], {t, 0, max}, {z, 0, max}];
coes = Transpose[ #*Range[0, max]! & /@ CoefficientList[se, {t, z}]]; Join[{0}, Flatten[ Table[ coes[[n, k]], {n, 2, max}, {k, 2, n-1}]]] (* From Jean-François Alcover, Oct 24 2011, after g.f. *)
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CROSSREFS
| Cf. A046740. Row sums give A000166. Diagonals give A070313, A070315.
A000274 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]
Sequence in context: A119727 A157272 A176200 * A056752 A053714 A168290
Adjacent sequences: A046736 A046737 A046738 * A046740 A046741 A046742
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KEYWORD
| nonn,easy,nice,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000
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