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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.
From Peter Bala, Mar 02 2011: (Start)
Companion triangles are A104035 and A185896.
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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G. C. Greubel, Rows n=0..101 of triangle, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
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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]
From Peter Bala, Feb 07 2011: (Start)
RELATION WITH BERNOULLI NUMBERS A000367 AND A002445
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
353792,0,11184128,0,71867136,0,191431680,0,250145280,0,159667200,0,39916800
0,22368256,0,309836800,0,1436058624,0,3149752320,0,3597834240,0,2075673600, 0,479001600
From Peter Bala, Feb 07 2011: (Start)
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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For other versions of this triangle see A008293, A101343.
A104035 is a companion triangle.
Highest order coefficients give factorials A000142. Constant terms give tangent numbers A000182. Other coefficients: A002301.
Setting u=1 in P_n gives A000831, u=2 gives A156073, u=3 gives A156075, u=4 gives A156076, u=1/2 gives A156102.
Setting u=sqrt(2) in P_n gives A156108 and A156122; setting u=sqrt(3) gives A156103 and A000436.
Cf. A008292, A019538, A185896.
Sequence in context: A252706 A139137 A138231 * A076880 A082115 A161553
Adjacent sequences: A155097 A155098 A155099 * A155101 A155102 A155103
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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N. J. A. Sloane, Nov 05 2009
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EXTENSIONS
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Name clarified by Peter Luschny, May 25 2015
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STATUS
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approved
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