cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109
Offset: 0

Views

Author

Vladeta Jovovic, Aug 31 2006

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

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

Programs

  • Maple
    a := n -> add(k^n*k!*combinat[stirling2](n,k),k=0..n); # Max Alekseyev, Feb 01 2007
  • Mathematica
    Flatten[{1,Table[Sum[k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)
  • PARI
    {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
  • PARI
    {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
  • PARI
    /* 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

Formula

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

Extensions

More terms from Max Alekseyev, Feb 01 2007

A220181 E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.

Original entry on oeis.org

1, 1, 7, 115, 3451, 164731, 11467387, 1096832395, 138027417451, 22111390122811, 4393756903239067, 1060590528331645675, 305686632592587314251, 103695663062502304228891, 40895823706632785802087547, 18554695374154504939196298955, 9596336362873294022956267703851
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2012

Keywords

Comments

Compare to the trivial identity: exp(x) = Sum_{n>=0} (1 - exp(-x))^n.
Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.
From Arvind Ayyer, Oct 25 2020: (Start)
a(n) is also the number of acyclic orientations with unique sink of the complete bipartite graph K_{n,n+1}
a(n) is also the number of toppleable permutations in S_{2n}. A toppleable permutation pi in S_{2n} satisfies pi_i <= n-1+i for 1 <= i <= n+1 and pi_i >= i-n for n+2 <= i <= 2n. (End)
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, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 ...], with an apparent period of 6. Cf. A122399. - Peter Bala, Jun 01 2022

Examples

			O.g.f.: F(x) = 1 + x + 7*x^2 + 115*x^3 + 3451*x^4 + 164731*x^5 +...
where F(x) = 1 + x/(1+x) + 2^2*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^3*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^4*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! +...
where the e.g.f. satisfies the identities:
(1) A(x) = 1 + (1-exp(-x)) + (1-exp(-2*x))^2 + (1-exp(-3*x))^3 + (1-exp(-4*x))^4 + (1-exp(-5*x))^5 + (1-exp(-6*x))^6 +...
(2) A(x) = exp(-x) + exp(-2*x)*(1-exp(-2*x)) + exp(-3*x)*(1-exp(-3*x))^2 + exp(-4*x)*(1-exp(-4*x))^3 + exp(-5*x)*(1-exp(-5*x))^4 + exp(-6*x)*(1-exp(-6*x))^5 +...
(3) 2*A(x) = 2 + (1-exp(-2*x)) + (1-exp(-3*x))^2 + (1-exp(-4*x))^3 + (1-exp(-5*x))^4 + (1-exp(-6*x))^5 + (1-exp(-7*x))^6 +...
E.g.f. at offset=1 begins:
B(x) = x + x^2/2! + 7*x^3/3! + 115*x^4/4! + 3451*x^5/5! + 164731*x^6/6! +...
where
B(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2^2 + (1-exp(-3*x))^3/3^2 + (1-exp(-4*x))^4/4^2 + (1-exp(-5*x))^5/5^2 + (1-exp(-6*x))^6/6^2 +...
The series  B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2  may be rewritten as:
B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{n>=2} (n-1)*exp(-2*n*x)/(2*n) -
Sum_{n>=3} C(n-1,2)*exp(-3*n*x)/(3*n) + Sum_{n>=4} C(n-1,3)*exp(-4*n*x)/(4*n) -+...

Links

Crossrefs

Cf. A136126, A092552, A048163, A220179, A122399, A187755, A203798, A320096, A338040.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[(-1)^(n-k)*k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)
  • PARI
    {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), ", "))
  • PARI
    {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Formula for this sequence with offset=1: */
{a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k*x+x*O(x^n)))^k/k^2), n)}
for(n=1, 21, print1(a(n), ", "))
  • PARI
    {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = sum(k=0,n,(-1)^(n-k)*k^n*k!*Stirling2(n, k))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,1,sum(k=1,n+1,((k-1)!)^2*Stirling2(n+1,k)^2/2))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0,n, k^n*sum(j=0,k, (-1)^(n+k-j)*binomial(k,j)*(k-j)^n))}
for(n=0, 20, print1(a(n), ", "))

Formula

O.g.f. Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 + n*k*x).
E.g.f. A(x) = Sum_{n>=0} (1 - exp(-n*x))^n satisfies the identities:
(1) A(x) = Sum_{n>=1} exp(-n*x) * (1 - exp(-n*x))^(n-1).
(2) A(x) = 1 + (1/2) * Sum_{n>=1} (1 - exp(-n*x))^(n-1).
(3) A(x) = Sum_{n>=1} Sum_{k>=0} (-1)^k * C(n+k-1,k) * exp(-k*(n+k-1)*x).
E.g.f. at offset 1, B(x) = Sum_{n>=1} a(n-1)*x^n/n!, satisfies:
(1) B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2.
(2) B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{k>=2} Sum_{n>=k} (-1)^k * C(n-1,k-1) * exp(-k*n*x)/(k*n), a convergent series for x>0.
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k).
a(n) = Sum_{k=1..n+1} ((k-1)!)^2 * Stirling2(n+1,k)^2 / 2 for n>0 with a(0)=1.
a(n) = Sum_{k=0..n} k^n * Sum_{j=0..k} (-1)^(n+k-j) * binomial(k,j) * (k-j)^n.
a(n) = A048163(n+1)/2 for n>0.
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... - Vaclav Kotesovec, Jun 21 2013
a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 13 2014

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

Views

Author

Paul D. Hanna, Jul 24 2013

Keywords

Comments

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

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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

A195005 E.g.f.: Sum_{n>=0} 2^n*(exp(n*x) - 1)^n.

Original entry on oeis.org

1, 2, 34, 1490, 122530, 16227602, 3155309794, 846406200530, 299510392317730, 135163342884412562, 75760096553546176354, 51633670624622762956370, 42049600429338786951232930, 40326932840083815683430101522, 44984263429111569097120217311714
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 34*x^2/2! + 1490*x^3/3! + 122530*x^4/4! +...
where
A(x) = 1 + 2*(exp(x)-1) + 2^2*(exp(2*x)-1)^2 + 2^3*(exp(3*x)-1)^3 +...

Links

Crossrefs

Cf. A195263, A122399, A301581, A338040, A338044.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[2^k * k^n * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Oct 04 2020 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n));n!*polcoeff(sum(m=0,n,2^m*(exp(m*X)-1)^m),n)}
  • PARI
    {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, 2^k*k^n*k!*Stirling2(n, k))}

Formula

a(n) = Sum_{k=0..n} 2^k*k^n*k!*Stirling2(n,k).
a(n) ~ c * (1 + 2*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.3559088366632706316517829481255877447669425726507348... - Vaclav Kotesovec, Oct 04 2020

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

Views

Author

Paul D. Hanna, Dec 31 2012

Keywords

Comments

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

Examples

			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 +...

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", ") )
  • PARI
    {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),", ") )

Formula

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

Views

Author

Paul D. Hanna, Dec 31 2012

Keywords

Comments

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

Examples

			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 +...

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", ") )

Formula

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

A245322 E.g.f.: Sum_{n>=0} sin(n*x)^n.

Original entry on oeis.org

1, 1, 8, 161, 6016, 360421, 31628288, 3823725821, 609263681536, 123729353398441, 31195066498285568, 9560281195915697081, 3500145542231863853056, 1508772905238685631514061, 756360258034794813559144448, 436312320288025061112662937941, 286966475921556619941746443288576
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 05 2014

Keywords

Comments

It appears that for n >= 1, a(2*n) is even and a(2*n-1) is odd. 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. A224899 and A221078. - Peter Bala, May 31 2022

Links

Crossrefs

Cf. A122399, A195415, A220181, A221077, A221078, A224899, A249489, A338040.

Programs

  • Mathematica
    nmax=20; Flatten[{1,Rest[CoefficientList[Series[Sum[Sin[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]}]
Flatten[{1,Table[Sum[(-1)^k * (n-2*k)^n * 2^(2*k-n) * Sum[Binomial[n-2*k,j] * (-1)^j * (n-2*k-2*j)^n,{j,0,n-2*k}],{k,0,n/2}],{n,1,20}]}]
  • PARI
    {a(n)=n!*polcoeff(sum(k=0, n, sin(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.6508143537621057095493599669955786931108630276472035393383790812849064745..., c = 0.447880926276318254580767843378566025547642779941081708311676940459098... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A195263 E.g.f.: Sum_{n>=0} 3^n*(exp(n*x) - 1)^n.

Original entry on oeis.org

1, 3, 75, 4809, 578415, 112024353, 31851411375, 12493267169169, 6464106627329055, 4265281191267407073, 3495556570494504442575, 3483310917470882398369329, 4147647341931988462919773695, 5815857702618060221437908948993, 9485411994735540168549266106329775
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 75*x^2/2! + 4809*x^3/3! + 578415*x^4/4! +...
where
A(x) = 1 + 3*(exp(x)-1) + 3^2*(exp(2*x)-1)^2 + 3^3*(exp(3*x)-1)^3 +...

Links

Crossrefs

Cf. A195005, A122399, A301582, A338040.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[3^k * k^n * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Oct 04 2020 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 3^m*(exp(m*X)-1)^m), n)}
  • PARI
    {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, 3^k*k^n*k!*Stirling2(n, k))}

Formula

a(n) = Sum_{k=0..n} 3^k*k^n*k!*Stirling2(n,k).
a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.36805443792839553744923868309093616341812244322234916... - Vaclav Kotesovec, Oct 04 2020

A195415 E.g.f.: Sum_{n>=1} tanh(n*x)^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

Views

Author

Paul D. Hanna, Sep 17 2011

Keywords

Comments

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

Examples

			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 +...

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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)}

Formula

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

A301583 G.f.: Sum_{n>=0} 4^n * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 4, 64, 1792, 70736, 3600128, 224255040, 16521605376, 1405131880000, 135480346104896, 14602769310474240, 1739917222954854400, 227081534040721917952, 32217108743091290851328, 4936803887495636263284736, 812576030237749532251019264, 142976863303365903802301729024, 26781577193841845859144244087808, 5320767287406003709062843236972544, 1117525692987087894816123931091214336
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Comments

In general, if k > 0 and g.f.: Sum_{j>=0} k^j * ((1+x)^j - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n! / 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). - Vaclav Kotesovec, Oct 08 2020

Examples

			G.f.: A(x) = 1 + 4*x + 64*x^2 + 1792*x^3 + 70736*x^4 + 3600128*x^5 + 224255040*x^6 + 16521605376*x^7 + 1405131880000*x^8 + ...
such that
A(x) = 1 + 4*((1+x)-1) + 16*((1+x)^2-1)^2 + 64*((1+x)^3-1)^3 + 256*((1+x)^4-1)^4 + 1024*((1+x)^5-1)^5 + 4096*((1+x)^6-1)^6 + ...
Also,
A(x) = 1/5 + 4*(1+x)/(1 + 4*(1+x))^2 + 16*(1+x)^4/(1 + 4*(1+x)^2)^3 + 64*(1+x)^9/(1 + 4*(1+x)^3)^4 + 256*(1+x)^16/(1 + 4*(1+x)^4)^5 + 1024*(1+x)^25/(1 + 4*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A122400, A301581, A301582, A301463.
Cf. A122399, A195005, A195263, A338040.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[4^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
  • PARI
    {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 4^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 4^n * (1+x)^(n^2) /(1 + 4*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 4*exp(1/r)) * r^2 = 11.35554580636894436474777793373210745006910386794268638744346793426715754570218..., 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 = 0.358692703763731594549618907599728117285634153... - Vaclav Kotesovec, Aug 09 2018, updated Oct 08 2020
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