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Search: id:A155100
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| A155100 |
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Triangle of coefficients in polynomials P_n(u) arising from the expansion of D^n (tan x) in increasing powers of tan x. |
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15
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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
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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The definition is d^n tan x / dx^n = P_n(tan x).
Interpolates between factorials and tangent numbers.
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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.
Haigh, Gordon; 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.
Knuth, D. E. and Buckholtz, Thomas J., Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.
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LINKS
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M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons
Michael E. Hoffman, DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES
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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]
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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
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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:
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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.
Sequence in context: A166065 A139137 A138231 this_sequence A076880 A082115 A161553
Adjacent sequences: A155097 A155098 A155099 this_sequence A155101 A155102 A155103
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KEYWORD
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nonn,tabl,new,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Nov 05 2009
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