A245322 -id:A245322 - cp's OEIS frontend

A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).

1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109

Offset: 0

Vladeta Jovovic, Aug 31 2006

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, 2, 1, 3, 3, 0, 1, 2, 1, 3, 3, 0, ...], with an apparent period of 6. Cf. A338040. - Peter Bala, May 31 2022

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! + ...
such that
A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 + ...
The e.g.f. is also given by the series:
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 + ...
or, equivalently,
A(x) = 1/2 + exp(-x)/(1+exp(-x))^2 + exp(-2*x)/(1+exp(-2*x))^3 + exp(-3*x)/(1+exp(-3*x))^4 + exp(-4*x)/(1+exp(-4*x))^5 + exp(-5*x)/(1+exp(-5*x))^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A195415, A122400, A220181, A224899, A245322, A320083, A338040.

a := n -> add(k^n*k!*combinat[stirling2](n,k),k=0..n); # Max Alekseyev, Feb 01 2007

Flatten[{1,Table[Sum[k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

{a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

{a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2014

/* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014

E.g.f.: Sum_{n >= 0} (exp(n*x) - 1)^n. - Vladeta Jovovic, Sep 03 2006
E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014
E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014
O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013
a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018
Let A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + ... denote the e.g.f. of the sequence. Let F(x) denote the series reversion of A(x) - 1 = x - 9*x^2/2 + 16*x^3/3 - 205*x^4/4 - 2714*x^5/5 - .... Then both dF/dx = 1 - 9*x + 16*x^2 - 205*x^3 - 2714*x^4 - ... and exp(F(x)) = 1 + x - 4*x^2 + x^3 - 38*x^4 - 606*x^5 - ... have integer coefficients. Note that 1 + series reversion(exp(F(x)) - 1) is the o.g.f. for A122400. - Peter Bala, Aug 09 2022

More terms from Max Alekseyev, Feb 01 2007

A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.

Original entry on oeis.org

1, 4, 132, 11140, 1763076, 449262724, 168055179012, 86720706877060, 59029852191779076, 51241585497612147844, 55245853646893977682692, 72423868722672448652558980, 113447698393867318106045295876, 209271794145089904620369489016964

Offset: 0

Vaclav Kotesovec, Oct 08 2020

nonn

In general, if k > 0 and e.g.f.: Sum_{j>=0} k^j * (exp(j*x) - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k).

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 [4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, ...], with an apparent period of 6. - Peter Bala, May 31 2022

Seiichi Manyama, Table of n, a(n) for n = 0..207

Cf. A122399, A195005, A195263, A195415, A220181, A221077, A221078, A224899, A245322.

Cf. A122400, A301581, A301582, A301583.

Flatten[{1, Table[Sum[4^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[4^j*(Exp[j*x] - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 4^m*(exp(m*X)-1)^m), n)}

a(n) = Sum_{j=0..n} 4^j * j^n * j! * Stirling2(n,j).

a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.37483929689722634406486945426531890297038414869116425498643733178324...

A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^n.

Original entry on oeis.org

1, 1, 8, 163, 6272, 389581, 35560448, 4479975823, 744707981312, 157897753198201, 41585725184933888, 13318468253704790683, 5097100004294081380352, 2297277197389011910783621, 1204339195916670860817072128, 726625952070893090583192860743

Offset: 0

Paul D. Hanna, Jul 24 2013

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, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 163*x^3/3! + 6272*x^4/4! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..225

Cf. A122399, A249489, A245322, A220181, A221077, A221078, A198513, A220181, A249459, A195415, A245322, A338040.

Flatten[{1,Table[Sum[Sum[Binomial[k,j] * (-1)^j * k^n*(k-2*j)^n / 2^k,{j,0,k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 29 2014 *)
Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Sinh[n*x]^n,{n,nn}],{x,0,nn}],x] Range[0,nn]!]]] (* Harvey P. Dale, May 18 2018 *)

{a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

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

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / (sqrt(3-2*log(2)) * 3^(n+1/2) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

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

Original entry on oeis.org

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

A195415 E.g.f.: Sum_{n>=1} tanh(nx)^n = Sum_{n>=1} a(n)4^(n-1)*x^n/n!.

Original entry on oeis.org

1, 2, 10, 92, 1351, 28982, 855100, 33214232, 1642999501, 100843185962, 7520379392890, 669760178257172, 70211429619908851, 8558006664633638942, 1200128210993564085880, 191861070874818576596912, 34685967730611200643509401, 7041037426518318365605795922

Offset: 1

Paul D. Hanna, Sep 17 2011

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, ...], with an apparent period of 6. - Peter Bala, May 29 2022

			E.g.f.: A(x) = x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
or, equivalently,
A(x) = x + 2*4*x^2/2! + 10*4^2*x^3/3! + 92*4^3*x^4/4! + 1351*4^4*x^5/5! +...
where
A(x) = 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 = 1..262

Cf. A122399, A220181, A221077, A221078, A224899, A245322, A338040.

seq(coeff(n!/4^(n-1)*series(add(tanh(n*x)^n, n = 1..100), x, 101), x, n), n = 1..100); # Peter Bala, May 29 2022

nmax = 20; Rest[CoefficientList[Series[Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! / 4^(Range[0, nmax] - 1)] (* Vaclav Kotesovec, May 31 2022 *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=1, n, tanh(m*X)^m); n!/4^(n-1)*polcoeff(Egf, n)}

E.g.f.: Sum_{n>=1} ( 1 - 2/(1+exp(2*n*x)) )^n = Sum_{n>=1} a(n)*4^(n-1)/n!.
a(n) ~ c * d^n * n^(2*n + 1/2), where d = 1 / (2 * exp(2) * log(1+sqrt(2))^2) = 0.0871085887239583895519632137900851584739951067757899616766024190... and c = 13.10490857177911562030370300610447966745088413236135355214718... - Vaclav Kotesovec, May 31 2022
a(n) = A221077(n) / 4^(n-1). - Vaclav Kotesovec, Jun 02 2022

A249489 a(n) = [x^n/n!] Sum_{k=0..n} cosh(k*x)^k.

Original entry on oeis.org

1, 0, 9, 0, 12070, 0, 126447741, 0, 5100496997940, 0, 562605048135059545, 0, 138523311740417986721274, 0, 66543520389763227261554370645, 0, 56664734898911130799849838608991176, 0, 79610326854782816434044397510470501877041

Offset: 0

Vaclav Kotesovec, Oct 30 2014

nonn

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

Cf. A224899, A249459, A245322, A221077, A221078.

Flatten[{1,Table[Sum[Sum[Binomial[k,j] * k^n*(k-2*j)^n / 2^k,{j,0,k}],{k,0,n}],{n,1,20}]}]

{a(n)=n!*polcoeff(sum(k=0, n, cosh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0,n,sum(j=0,k, binomial(k, j) * k^n*(k-2*j)^n / 2^k ))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 15 2018, using Vaclav's formula.

a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(k, j) * k^n * (k-2*j)^n / 2^k. [Explicitly stating Vaclav's formula in Mma program - Paul D. Hanna, Oct 15 2018]

If n is even, then a(n) ~ c * (1-2*r)^n * n^(2*n) / (2^n * exp(n) * (r*(1-r))^(n/2)), where r = 0.0832217201995176507819192648878903254298041... is the root of the equation (r/(1-r))^(1-2*r) = exp(-2), and c = 2.09233700490262732901066903251002074102409436600891921766318742438...

Name clarified by Paul D. Hanna, Oct 15 2018

cp's OEIS Frontend

A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).

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A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.

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A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^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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A195415 E.g.f.: Sum_{n>=1} tanh(n*x)^n = Sum_{n>=1} a(n)*4^(n-1)*x^n/n!.

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A249489 a(n) = [x^n/n!] Sum_{k=0..n} cosh(k*x)^k.

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A195415 E.g.f.: Sum_{n>=1} tanh(nx)^n = Sum_{n>=1} a(n)4^(n-1)*x^n/n!.