A085466 -id:A085466 - cp's OEIS frontend

A138729 a(n) = -A085466(n) times the free coefficient of the 2n-th Du Bois-Reymond polynomial in e^2.

7, 25, 98, 1167, 4650, 30930, 1111860, 31100895, 13812610, 894570642, 89419472100, 1311049104750, 47185076099700, 350440263072900, 4578242813103960, 4119778157653533375, 16476927824724617250, 124478128839848033250, 40326914169455544130500

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a = {7}; Do[p = FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x, I}]]]; AppendTo[a, -First[p[[2]]]], {n, 2, 9}]; a (*Artur Jasinski*)

Edited and extended by Max Alekseyev, Sep 15 2009

A085467 Numerators of coefficients of e^2 in the table of (2n)th du Bois Reymond constants.

Original entry on oeis.org

0, 1, -7, 1, -4, -25, 1, -6, 3, -98, 3, -24, 36, -8, -1167, 3, -30, 75, -40, -10, -4650, 5, -60, 210, -220, 35, -56, -30930, 45, -630, 2835, -4620, 2205, -378, -1589, -1111860, 315, -5040, 27720, -62160, 52500, -13104, -4424, -36304, -31100895, 35, -630, 4095, -11760, 14700, -6888

Offset: 0

Eric W. Weisstein, Jul 01 2003

			{(-7 + e^2)/2, (-25 - 4*e^2 + e^4)/8, (-98 + 3*e^2 - 6*e^4 + e^6)/32}.

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants

Cf. A085466.

c[2] = (1/2)*(E^2 - 7); c[n_] := Simplify[ ExpToTrig[ -2*Residue[ x^2/((x^2 + 1)^(n/2)*(Tan[x] - x)), {x, I}] - 3]]; row[n_] := Reverse[ coes = CoefficientList[ c[n], E^2]; coes*LCM @@ Denominator[coes]]; Flatten[ Table[ row[2*n], {n, 1, 9}]] (* Jean-François Alcover, Oct 25 2012, from formula given in A085466 *)

Sequence has been prepended with a(0)=0 to enable table display (so offset has been set to 0 accordingly) by Michel Marcus, Sep 11 2013

A138730 Continued fraction for 4th Du Bois Reymond constant.

Original entry on oeis.org

0, 190, 1, 4, 2, 1, 1, 1, 6, 10, 6, 6, 1, 3, 2, 9, 67, 2, 3, 1, 7, 1, 2, 1, 1, 1, 2, 1, 1, 4, 1, 68, 5, 6, 1, 1, 1, 4, 1, 1, 5, 1, 4, 8, 2, 5, 5, 1, 1, 3, 1, 2, 2, 6, 1, 1, 1, 9, 2, 1, 1, 1, 17, 5, 21, 2, 9, 1, 3, 1, 2, 4, 1, 3, 5, 1, 56, 1, 4, 14, 5, 17, 4, 2, 34, 1, 18, 1, 1, 4, 1, 5, 16, 1, 3, 27, 1, 11

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A207528, A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^2), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(4)-4*exp(2)-25)/8) \\ Charles R Greathouse IV, Feb 24 2012

A138731 Continued fraction for 6th Du Bois Reymond constant.

Original entry on oeis.org

0, 4531, 1, 1, 3, 1, 8, 3, 2, 1, 3, 1, 1, 9, 1, 2, 9, 1, 1, 5, 3, 2, 1, 1, 5, 1, 1, 3, 5, 3, 2, 1, 6, 1, 7, 2, 4, 1, 3, 1, 23, 2, 8, 2, 1, 19, 1, 23, 1, 3, 1, 18, 1, 2, 1, 1, 3, 24, 1, 4, 1, 1, 5, 1, 1, 2, 1, 1, 89, 1, 1, 1, 2, 1, 8, 1, 52, 1, 1, 1, 1, 1, 4, 1, 15, 1, 1, 1, 5, 3, 1, 2, 1, 4, 1, 4, 24, 5, 1, 5

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^3), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(6)-6*exp(4)+3*exp(2)-98)/32) \\ Charles R Greathouse IV, Feb 24 2012

A138732 Continued fraction for 8th Du Bois Reymond constant.

Original entry on oeis.org

0, 99232, 2, 6, 1, 3, 4, 2, 1, 2, 1, 16, 8, 2, 57, 13, 2, 1, 16, 1, 1, 6, 5, 5, 1, 1, 6, 1, 1, 3, 1, 1, 2, 9, 18, 1, 8, 15, 2, 2, 1, 2, 1, 2, 1, 5, 2, 3, 5, 1, 3, 4, 17, 11, 1, 2, 1, 6, 1, 2, 3, 15, 3, 12, 1, 8, 6, 1, 1, 1, 2, 4, 29, 44, 1, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 22, 1, 2, 5, 8, 2, 2, 5, 2

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^4), {x, I}]]], 100] (* Artur Jasinski *)

contfrac((3*exp(8) - 24*exp(6) + 36*exp(4) - 8*exp(2)  - 1167)/384) \\ Michel Marcus, Sep 09 2013

A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

Original entry on oeis.org

5, 190, 4531, 99232, 2125044, 45190209, 958768567, 20325471335, 430773893366, 9128872855695, 193450867955197, 4099389985205820, 86869246502331992, 1840823999333339814, 39008411877876819180, 826616742911186406242

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a = {}; Do[AppendTo[a,Last[ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x,I}]]], 2]]], {n, 1, 9}]; a (*Artur Jasinski*)

a(n) = floor(1/C(2n)), where C(2n) is the 2n-th Du Bois Reymond constant. [From Max Alekseyev, Sep 15 2009]

Extended by Max Alekseyev, Sep 15 2009

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

Original entry on oeis.org

0, 1, 0, 1, -1, -1, -1, 0, 0, 3, 1, -5, 18, -13, -7, -11, 70, -135, 65, -10, 45, 111, -609, 1215, -1350, 1275, -621, -141, -1009, 6188, -16758, 27335, -26845, 12474, -2548, 1883, 10977, -81353, 270004, -511791, 584710, -420287, 216468, -70169, -3599, -146691, 1248210, -4715217, 10303461, -14439411

Offset: 0

Gerald McGarvey, Mar 02 2005

For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].

Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.

Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]

For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).

For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).

Added the keyword tabl Gerald McGarvey, Aug 20 2009

cp's OEIS Frontend

A138729 a(n) = -A085466(n) times the free coefficient of the 2n-th Du Bois-Reymond polynomial in e^2.

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A085467 Numerators of coefficients of e^2 in the table of (2n)th du Bois Reymond constants.

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A138730 Continued fraction for 4th Du Bois Reymond constant.

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A138731 Continued fraction for 6th Du Bois Reymond constant.

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A138732 Continued fraction for 8th Du Bois Reymond constant.

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A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).

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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).