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A155100
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Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0.
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17
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1, 0, 1, 1, 0, 1, 0, 2, 0, 2, 2, 0, 8, 0, 6, 0, 16, 0, 40, 0, 24, 16, 0, 136, 0, 240, 0, 120, 0, 272, 0, 1232, 0, 1680, 0, 720, 272, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 7936, 0, 176896, 0, 814080, 0, 1491840
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OFFSET
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0,8
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COMMENTS
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The definition is d^(n-1) tan x / dx^n = P_n(tan x) for n>=1 and 1 for n=0.
Interpolates between factorials and tangent numbers.
A combinatorial interpretation for the polynomial P_n(t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...,|x_n|} = {1,2,...,n}.
They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation and put x_0 = -(n+1) and x_(n+1) = (-1)^n*(n+1). Then x_0,x_1,...,x_n,x_(n+1) is a snake of type S(n) when x_0 < x_1 > x_2 < ... x_(n+1). For example, -5 4 -3 -1 -2 5 is a snake of type S(4).
Let sc be the number of sign changes through a snake sc = #{i, 0 <= i <= n, x_i*x_(i+1) < 0}. For example, the snake -5 4 -3 -1 -2 5 has sc = 3.
The polynomial P_(n+1)(t) is the generating function for the sign change statistic on snakes of type S(n): P_(n+1)(t) = sum {snakes in S(n)} t^sc.
See the example section below for the cases n=1 and n=2.
(End)
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
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LINKS
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FORMULA
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If the polynomials are denoted by P_n(u), we have the recurrence P_{-1}=1, P_0 = u, P_n = (u^2+1)*dP_{n-1}/du.
G.f.: Sum_{n >= 0} P_n(u) t^n/n! = (sin t + u*cos t)/(cos t - u sin t). [Hoffman]
Put T(n,t) = P_n(i*t), where i = sqrt(-1). We have the definite integral evaluation, valid when both m and n are >=1 and m+n >= 4:
int( T(m,t)*T(n,t)/(1-t^2), t = -1..1) = (-1)^((m-n)/2)*2^(m+n-1)*Bernoulli(m+n-2).
The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
RELATION WITH OTHER ROW POLYNOMIALS
The following three identities hold for n >= 1:
P_(n+1)(t) = (1+t^2)*R(n-1,t) where R(n,t) is the n-th row polynomial of A185896.
P_(n+1)(t) = (-2*i)^n*(t-i)*R(n,-1/2+1/2*i*t), where i = sqrt(-1) and R(n,x) is an ordered Bell polynomial, that is, the n-th row polynomial of A019538.
P_(n+1)(t) = (t-i)*(t+i)^n*A(n,(t-i)/(t+i)), where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials - see A008292. (End)
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EXAMPLE
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The polynomials P_{-1}(u) through P_6(u) with exponents in decreasing order:
1
u
u^2 + 1
2*u^3 + 2*u
6*u^4 + 8*u^2 + 2
24*u^5 + 40*u^3 + 16*u
120*u^6 + 240*u^4 + 136*u^2 + 16
720*u^7 + 1680*u^5 + 1232*u^3 + 272*u
...
Triangle begins:
1
0, 1
1, 0, 1
0, 2, 0, 2
2, 0, 8, 0, 6
0, 16, 0, 40, 0, 24
16, 0, 136, 0, 240, 0, 120
0, 272, 0, 1232, 0, 1680, 0, 720
272, 0, 3968, 0, 12096, 0, 13440, 0, 5040
0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320
7936, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880
0, 353792, 0, 3610112, 0, 12207360, 0, 18627840, 0, 13305600, 0, 3628800
...
Examples of sign change statistic sc on snakes of type S(n):
Snakes # sign changes sc t^sc
=========== ================= ====
n=1:
-2 1 -2 ........... 2 ........ t^2
-2 -1 -2 ........... 0 ........ 1
yields P_2(t) = 1 + t^2;
n=2:
-3 1 -2 3 ........ 3 ........ t^3
-3 2 1 3 ........ 1 ........ t
-3 2 -1 3 ........ 3 ........ t^3
-3 -1 -2 3 ........ 1 ........ t
yields P_3(t) = 2*t + 2*t^3. (End)
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MAPLE
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P:=proc(n) option remember;
if n=-1 then RETURN(1); elif n=0 then RETURN(u); else RETURN(expand((u^2+1)*diff(P(n-1), u))); fi;
end;
for n from -1 to 12 do t1:=series(P(n), u, 20); lprint(seriestolist(t1)); od:
# Alternatively:
with(PolynomialTools): seq(print(CoefficientList(`if`(i=0, 1, D@@(i-1))(tan), tan)), i=0..7); # Peter Luschny, May 19 2015
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MATHEMATICA
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p[n_, u_] := D[Tan[x], {x, n}] /. Tan[x] -> u /. Sec[x] -> Sqrt[1 + u^2] // Expand; p[-1, u_] = 1; Flatten[ Table[ CoefficientList[ p[n, u], u], {n, -1, 9}]] (* Jean-François Alcover, Jun 28 2012 *)
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CROSSREFS
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Highest order coefficients give factorials A000142. Constant terms give tangent numbers A000182. Other coefficients: A002301.
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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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