A258927 -id:A258927 - cp's OEIS frontend

A258880 E.g.f. satisfies: A(x) = Integral 1 + A(x)^3 dx.

1, 6, 540, 184680, 157600080, 270419925600, 816984611467200, 3971317527112003200, 29097143353353192480000, 305823675529741700675520000, 4435486895868663971869188480000, 86036822683997062842122964537600000, 2175352015640142857526698650779456000000

Offset: 0

Paul D. Hanna, Jun 13 2015

nonn

Note: Sum_{n>=0} (-1)^n*x^(3*n+1)/(3*n+1) = log( (1+x)/(1-x^3)^(1/3) )/2 + Pi*sqrt(3)/18 - atan( (1-2*x)*sqrt(3)/3 )*sqrt(3)/3.

			E.g.f.: A(x) = x + 6*x^4/4! + 540*x^7/7! + 184680*x^10/10! + 157600080*x^13/13! + 270419925600*x^16/16! +...
where Series_Reversion(A(x)) =  x - x^4/4 + x^7/7 - x^10/10 + x^13/13 - x^16/16 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150
Guo-Niu Han, Jing-Yi Liu, Divisibility properties of the tangent numbers and its generalizations, arXiv:1707.08882 [math.CO], 2017. See Table for k = 3 p. 8.

Cf. A000182, A000831, A258878, A258901, A258925, A258927, A259112, A259113, A258969.

terms = 13;
A[_] = 0;
Do[A[x_] = Integrate[1 + A[x]^3, x] + O[x]^k // Normal, {k, 1, 3 terms}];
DeleteCases[CoefficientList[A[x], x] Range[0, 3 terms - 2]!, 0] (* Jean-François Alcover, Jul 25 2018 *)

{a(n) = local(A=x); A = serreverse( sum(m=0,n, (-1)^m * x^(3*m+1)/(3*m+1) ) +O(x^(3*n+2)) ); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))

/* E.g.f. A(x) = Integral 1 + A(x)^3 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^3 + O(x^(3*n+2)) )); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))

E.g.f.: Series_Reversion( Integral 1/(1+x^3) dx ).
E.g.f.: Series_Reversion( Sum_{n>=0} (-1)^n * x^(3*n+1)/(3*n+1) ).
a(n) ~ 3^(15*n/2 + 17/4) * n^(3*n+1) / (exp(3*n) * (2*Pi)^(3*n+3/2)). - Vaclav Kotesovec, Jun 15 2015

A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

Original entry on oeis.org

1, 24, 32256, 285272064, 8967114326016, 735868743566229504, 130778914961055994085376, 44390350317502907443360825344, 26290393222157669992962395876622336, 25377887922329300948014930852183837507584, 37855568618678541873143615775486954119570128896

Offset: 0

Paul D. Hanna, Jun 14 2015

nonn

			E.g.f.: A(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! + 735868743566229504*x^21/21! +...
where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! - 8310698364852576*x^21/21! +...+ A258900(n)*x^(4*n+1)/(4*n+1)! +...
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! - 8310698364852576*x^20/20! +...+ A258900(n)*x^(4*n)/(4*n)! +...
such that C(x)^4 + S(x)^4 = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..112
Guo-Niu Han, Jing-Yi Liu, Divisibility properties of the tangent numbers and its generalizations, arXiv:1707.08882 [math.CO], 2017. See Table for k = 4 p. 8.

Cf. A000182, A000831, A258900, A258880, A258925, A258927, A259112, A259113, A258970.

nmax=20; Table[CoefficientList[InverseSeries[Series[Integrate[1/(1+x^4),x],{x,0,4*nmax+1}],x],x][[4*n-2]] * (4*n-3)!, {n,1,nmax+1}] (* Vaclav Kotesovec, Jun 18 2015 *)

/* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^4 + O(x^(4*n+2))) ) ); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

/* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^4 + O(x^(4*n+2)) )); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^4) dx ).
(2) A(x) = sqrt( tan( 2 * Integral A(x) dx ) ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ) is the e.g.f. of A258900, then e.g.f. A(x) of this sequence satisfies:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^4 dx,
(5) A(x)^2 = S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
a(n) ~ 2^(6*n + 14/3) * (4*n)! * n^(1/3) / (3^(1/3) * Gamma(1/3) * Pi^(4*n + 4/3)). - Vaclav Kotesovec, Jun 18 2015

A258925 E.g.f. satisfies: A(x) = Integral 1 + A(x)^5 dx.

Original entry on oeis.org

1, 120, 3024000, 858574080000, 1226178516326400000, 5912338932461445120000000, 75732595735526211882516480000000, 2195068320271703663798288449536000000000, 128322069958974226301129597680106864640000000000

Offset: 0

Paul D. Hanna, Jun 15 2015

nonn

			E.g.f.: A(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! +...
where Series_Reversion(A(x)) = x - x^6/6 + x^11/11 - x^16/16 + x^21/21 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +...+ A258924(n)*x^(5*n+1)/(5*n+1)! +...
C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +...+ A258924(n)*x^(5*n)/(5*n)! +...
such that C(x)^5 + S(x)^5 = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..90

Cf. A258924, A258880, A258901, A258927, A259112, A259113, A258971.

/* E.g.f. Series_Reversion( Integral 1/(1+x^5) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^5 + O(x^(5*n+2))) ) ); (5*n+1)!*polcoeff(A, 5*n+1)}
for(n=0, 20, print1(a(n), ", "))

/* E.g.f. A(x) = Integral 1 + A(x)^5 dx.: */
{a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^5 + O(x^(5*n+2)) )); (5*n+1)!*polcoeff(A, 5*n+1)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^5) dx ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ) is the e.g.f. of A258924, then e.g.f. A(x) of this sequence satisfies:
(2) A(x) = S(x)/C(x),
(3) A(x) = Integral 1/C(x)^5 dx.
From Vaclav Kotesovec, Jun 18 2015: (Start)
a(n) ~ 5^(5*n+5/2) * ((5-sqrt(5))/8)^(5*n/2+5/8) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
a(n) ~ 5^(5*n+5/2) * sin(Pi/5)^(5*n+5/4) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
(End)

A258994 E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1.

Original entry on oeis.org

1, 2, 12, 192, 4272, 124992, 4531392, 195869952, 9832326912, 562125837312, 36056880110592, 2564230500421632, 200237330428342272, 17032391106795159552, 1567547894591436275712, 155196096043697480466432, 16447362605632117421309952, 1857733260790463501532659712

Offset: 0

Vaclav Kotesovec, Jun 16 2015

nonn

In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).

			A(x) = 1 + 2*x + 12*x^2/2! + 192*x^3/3! + 4272*x^4/4! + 124992*x^5/5! + ...
A'(x) = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
1 + A(x)^6 = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..50

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258971 (k=5), A258927.

nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k,{k,0,nmax+1}]; Table[Subscript[a,k]*k!,{k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^6-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]

{a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^6 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

a(n) ~ n! * d^(n+1/5) / (5^(1/5) * Gamma(1/5) * n^(4/5)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(6*j-1) = 6/(Pi - sqrt(3)*log(2+sqrt(3))) = 6.97224737278326506475991855023425659249063565...

E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^6) dx ). - Paul D. Hanna, Jun 16 2015

A259112 E.g.f. satisfies: A(x) = Integral 1 + A(x)^7 dx.

Original entry on oeis.org

1, 5040, 76281004800, 37626350120206848000, 185657801986983855655526400000, 5150422429203073500358041285476352000000, 569512147150397429576160463881863910421954560000000, 199607288101583292042564550150623446229209414764068864000000000

Offset: 0

Vaclav Kotesovec, Jun 18 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..64

Cf. A258880 (k=3), A258901 (k=4), A258925 (k=5), A258927 (k=6), A259113 (k=8).

{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^7 + O(x^(7*n+2))) ) ); (7*n+1)!*polcoeff(A, 7*n+1)}
for(n=0, 20, print1(a(n), ", ")) \\ after Paul D. Hanna

a(n) ~ 7^(7*n+7/3) * n^(1/6) * (sin(Pi/7)/Pi)^(7*n+7/6) * (7*n)! / (6^(1/6) * Gamma(1/6)).

A259113 E.g.f. satisfies: A(x) = Integral 1 + A(x)^8 dx.

Original entry on oeis.org

1, 40320, 18598035456000, 474009962689446543360000, 170149872975531014630262649651200000, 442695618409212548301531680485487369256960000000, 5620045472937667963281036681944526735620775198955929600000000

Offset: 0

Vaclav Kotesovec, Jun 18 2015

nonn

In general, for k>2, if e.g.f. satisfies A(x) = Integral 1 + A(x)^k dx, then a(n) ~ k^(k/(k-1)) * n^(1/(k-1)) * (k*n)! * (k*sin(Pi/k)/Pi)^(k*n + k/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1))).

Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Cf. A258880 (k=3), A258901 (k=4), A258925 (k=5), A258927 (k=6), A259112 (k=7).

{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^8 + O(x^(8*n+2))) ) ); (8*n+1)!*polcoeff(A, 8*n+1)}
for(n=0, 20, print1(a(n), ", ")) \\ after Paul D. Hanna

a(n) ~ 2^(24*n+48/7) * n^(1/7) * (sin(Pi/8)/Pi)^(8*n+8/7) * (8*n)! / (7^(1/7) * GAMMA(1/7)).

a(n) ~ 2^(16*n+40/7) * (2-sqrt(2))^(4*n+4/7) * n^(1/7) * (8*n)! / (7^(1/7) * GAMMA(1/7) * Pi^(8*n+8/7)).

A258924 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, -24, -169344, -25255286784, -23089632627769344, -79051067969864491597824, -766667475511149432871084621824, -17578325209217134578862801556544159744, -839197248407269659950832532302025663168118784

Offset: 0

Paul D. Hanna, Jun 15 2015

sign

			E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:
C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +...
S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +...
such that C(x)^5 + S(x)^5 = 1:
C(x)^5 = 1 - 120*x^5/5! + 604800*x^10/10! + 13208832000*x^15/15! +...
S(x)^5 = 120*x^5/5! - 604800*x^10/10! - 13208832000*x^15/15! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^5)^(1/5) dx:
Series_Reversion(S(x)) = x + 24*x^6/6! + 435456*x^11/11! + 115075344384*x^16/16! +...
1/(1-x^5)^(1/5) = 1 + 24*x^5/5! + 435456*x^10/10! + 115075344384*x^15/15! +...
(2) d/dx S(x)/C(x) = 1/C(x)^5:
1/C(x)^5 = 1 + 120*x^5/5! + 3024000*x^10/10! + 858574080000*x^15/15! +...
S(x)/C(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! + 1226178516326400000*x^21/21! +...+ A258925(n)*x^(5*n+1)/(5*n+1)! +...
where
Series_Reversion(S(x)/C(x)) = x - 1/6*x^6 + 1/11*x^11 - 1/16*x^16 + 1/21*x^21 +...

Cf. A258925 (S/C), A258878, A258900, A258927.

/* E.g.f. Series_Reversion(Integral 1/(1-x^5)^(1/5) dx): */
{a(n)=local(S=x); S = serreverse( intformal(  1/(1-x^5 +x*O(x^(5*n)))^(1/5) )); (5*n+1)!*polcoeff(S, 5*n+1)}
for(n=0, 15, print1(a(n), ", "))

/* E.g.f. C(x) with offset 0: */
{a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(5*n))); C=1-intformal(S^4/C^3 +x*O(x^(5*n))); ); (5*n)!*polcoeff(C, 5*n)}
for(n=0, 15, print1(a(n), ", "))

/* E.g.f. S(x) with offset 1: */
{a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(5*n+1))); C=1-intformal(S^4/C^3 +x*O(x^(5*n+1))); ); (5*n+1)!*polcoeff(S, 5*n+1)}
for(n=0, 15, print1(a(n), ", "))

Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(5*n)/(5*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(5*n+1)/(5*n+1)!, then C(x) and S(x) satisfy:
(1) C(x)^5 + S(x)^5 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^4/C(x)^3,
(4) C(x)^4 * C'(x) + S(x)^4 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^5 dx,
(6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^5) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(5*n+1)/(5*n+1) ).

A258926 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^6)^(1/6) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, -120, -21859200, -131273353728000, -6725237593471119360000, -1653993087378574357912780800000, -1405832822961504544259161592168448000000, -3334380558587161259470375739654344298987520000000, -18982929854690021819576777610944622891185796965990400000000

Offset: 0

Paul D. Hanna, Jun 14 2015

sign

			E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:
C(x) = 1 - 120*x^6/6! - 21859200*x^12/12! - 131273353728000*x^18/18! -...
S(x) = x - 120*x^7/7! - 21859200*x^13/13! - 131273353728000*x^19/19! -...
such that C(x)^6 + S(x)^6 = 1:
C(x)^6 = 1 - 720*x^6/6! + 68428800*x^12/12! + 80406577152000*x^18/18! +...
S(x)^6 = 720*x^6/6! - 68428800*x^12/12! - 80406577152000*x^18/18! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^6)^(1/6) dx:
Series_Reversion(S(x)) = x + 120*x^7/7! + 46569600*x^13/13! + 449549388288000*x^19/19! +...
1/(1-x^6)^(1/6) = 1 + 120*x^6/6! + 46569600*x^12/12! + 449549388288000*x^18/18! +...
(2) d/dx S(x)/C(x) = 1/C(x)^6:
1/C(x)^6 = 1 + 720*x^6/6! + 410572800*x^12/12! + 4492717498368000*x^18/18! +...
S(x)/C(x) = x + 720*x^7/7! + 410572800*x^13/13! + 4492717498368000*x^19/19! + 348990783113936240640000*x^25/25! +...+ A258927(n)*x^(6*n+1)/(6*n+1)! +...
where
Series_Reversion(S(x)/C(x)) = x - x^7/7 + x^13/13 - x^19/19 + x^25/25 - x^31/31 +...

Cf. A258927 (S/C), A258878, A258900, A258924.

nmax = 8; a[n_] := SeriesCoefficient[ InverseSeries[ Integrate[1/(1 - x^6)^(1/6), x] + O[x]^(6nmax+2), x], 6n+1]*(6n+1)!; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 26 2017 *)

/* E.g.f. Series_Reversion(Integral 1/(1-x^6)^(1/6) dx): */
{a(n)=local(S=x); S = serreverse( intformal(  1/(1-x^6 +x*O(x^(6*n)))^(1/6) )); (6*n+1)!*polcoeff(S, 6*n+1)}
for(n=0, 15, print1(a(n), ", "))

/* E.g.f. C(x) with offset 0: */
{a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(6*n))); C=1-intformal(S^5/C^4 +x*O(x^(6*n))); ); (6*n)!*polcoeff(C,6*n)}
for(n=0, 21, print1(a(n), ", "))

/* E.g.f. S(x) with offset 1: */
{a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(6*n+1))); C=1-intformal(S^5/C^4 +x*O(x^(6*n+1))); ); (6*n+1)!*polcoeff(S,6*n+1)}
for(n=0, 21, print1(a(n), ", "))

Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(6*n)/(6*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(6*n+1)/(6*n+1)!, then C(x) and S(x) satisfy:
(1) C(x)^6 + S(x)^6 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^5/C(x)^4,
(4) C(x)^5 * C'(x) + S(x)^5 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^6 dx,
(6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^6) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(6*n+1)/(6*n+1) ).
(7) S(x)^3/C(x)^3 = tan( 3 * Integral S(x)^2/C(x)^2 dx ).
(8) C(x)^3 + I*S(x)^3 = exp( 3*I * Integral S(x)^2/C(x)^2 dx ).

cp's OEIS Frontend

A258880 E.g.f. satisfies: A(x) = Integral 1 + A(x)^3 dx.

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A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

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A258925 E.g.f. satisfies: A(x) = Integral 1 + A(x)^5 dx.

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PARI

PARI

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A258994 E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1.

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Mathematica

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A259112 E.g.f. satisfies: A(x) = Integral 1 + A(x)^7 dx.

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A259113 E.g.f. satisfies: A(x) = Integral 1 + A(x)^8 dx.

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A258924 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ), where the constant of integration is zero.

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PARI

PARI

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A258926 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^6)^(1/6) dx ), where the constant of integration is zero.