A221079 -id:A221079 - cp's OEIS frontend

A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

1, 1, 8, 160, 5888, 345856, 29677568, 3502489600, 544181977088, 107675615297536, 26435436140822528, 7885689342279024640, 2809177794704769548288, 1177952320402008693538816, 574318105367992485583781888, 322156963576521588458420961280, 206009256195720974104252003647488

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
where
A(x) = 1 + tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Cf. A122399, A195415, A220181, A221078, A221079, A224899, A249489, A245322, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tanh(m*X)^m); n!*polcoeff(Egf, n)}
for(n=0,20,print1(a(n),", ") )

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

E.g.f.: Sum_{n>=0} (exp(2*n*x) - 1)^n / (exp(2*n*x) + 1)^n.

a(n) ~ c * 2^n * (n!)^2 / (sqrt(n) * (log(1+sqrt(2)))^(2*n)), where c = 0.521427744491499132141002572969819345522922990165233786929882335275903215... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

Original entry on oeis.org

1, 1, 8, 164, 6400, 404176, 37541888, 4814990144, 815074508800, 176018678814976, 47223034903789568, 15407438848482919424, 6007522256082907955200, 2758698201106509138251776, 1473586749521302260021198848, 905915791153129699969076117504

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A245322. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 164*x^3/3! + 6400*x^4/4! + 404176*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(2*x)^2 + tan(3*x)^3 + tan(4*x)^4 + tan(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..224

Cf. A122399, A195415, A220181, A221077, A221079, A245322, A224899, A249489, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tan[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[(-1)^((n-k)/2) * 2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tan(m*X)^m); n!*polcoeff(Egf, n)}
for(n=0,20,print1(a(n),", ") )

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.82830192319144609189890882712268369027077465204866199572119508594067235975..., c = 0.3460492649810724519960613805096579760009441161242336020188358769124140... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A210526 E.g.f.: Sum_{n>=0} tan(n*x)^n / n!.

Original entry on oeis.org

1, 1, 4, 29, 384, 8001, 219200, 7639757, 338030592, 18221107681, 1161865470976, 86954156945501, 7533144553783296, 744010437498030561, 83107197798886031360, 10418926401222040969421, 1453531583873609883451392, 224285068607398484964201793, 38075052899686318527662522368

Offset: 0

Paul D. Hanna, Jan 27 2013

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 384*x^4/4! + 8001*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(2*x)^2/2! + tan(3*x)^3/3! + tan(4*x)^4/4! + tan(5*x)^5/5! + tan(6*x)^6/6! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A198513, A221079.

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tan(m*X)^m/m!); n!*polcoeff(Egf, n)}
for(n=0, 20, print1(a(n), ", "))

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A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

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A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

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A210526 E.g.f.: Sum_{n>=0} tan(n*x)^n / n!.

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PARI