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.

Showing 1-6 of 6 results.

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

Views

Author

Vaclav Kotesovec, Oct 08 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A122399, A195005, A195263, A195415, A220181, A221077, A221078, A224899, A245322.
Cf. A122400, A301581, A301582, A301583.

Programs

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

Formula

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...

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

A301582 G.f.: Sum_{n>=0} 3^n * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 3, 36, 765, 22932, 886707, 41971041, 2349915543, 151893243711, 11131097539221, 911906584505874, 82586031357156975, 8192750710914222984, 883506535094875209327, 102907862475072248379060, 12875067336646598300376165, 1722014444866824121524712497, 245185575019136812676809863351, 37027348593726417935247243009495, 5911490521308027393188499233189367, 994821814352463817234026392636083551
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 36*x^2 + 765*x^3 + 22932*x^4 + 886707*x^5 + 41971041*x^6 + 2349915543*x^7 + 151893243711*x^8 + ...
such that
A(x) = 1 + 3*((1+x)-1) + 9*((1+x)^2-1)^2 + 27*((1+x)^3-1)^3 + 81*((1+x)^4-1)^4 + 243*((1+x)^5-1)^5 + 729*((1+x)^6-1)^6 + 2187*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/4 + 3*(1+x)/(1 + 3*(1+x))^2 + 9*(1+x)^4/(1 + 3*(1+x)^2)^3 + 27*(1+x)^9/(1 + 3*(1+x)^3)^4 + 81*(1+x)^16/(1 + 3*(1+x)^4)^5 + 243*(1+x)^25/(1 + 3*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A122400, A195263, A301581, A301583, A301463.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[3^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, 3^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 3^n * (1+x)^(n^2) /(1 + 3*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 3*exp(1/r)) * r^2 = 8.632012704198046828204904686098781240870113556702123911346365466059061495897353..., 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 = 0.34734097623709084937300542950550592394946492732014... - Vaclav Kotesovec, Aug 09 2018

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

A350722 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 33, 4567, 1652493, 1235777551, 1656820330173, 3619858882041487, 12034209740498292093, 57813156798714532953391, 385490564193781368103929213, 3454086424032897924417605526607, 40500898779980258599522326286912893
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Crossrefs

Cf. A122399, A195005, A195263, A338040.
Cf. A108459, A350721, A351117.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^(k+n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*k^(k+n)*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k)))

Formula

E.g.f.: Sum_{k>=0} (k * (exp(k*x) - 1))^k.
a(n) ~ exp(exp(-2)/2) * n! * n^(2*n). - Vaclav Kotesovec, Feb 04 2022

A350720 a(n) = Sum_{k=0..n} k! * 3^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 3, 69, 3948, 422082, 72567522, 18304992558, 6367730357160, 2921446409138136, 1709074810258369776, 1241694104839498851552, 1096850187800368469477424, 1157691464039682741551221152, 1438880771284303822650674399664
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A320083, A350719.
Cf. A195263, A335531, A350721.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 3^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*3^k*k^n*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*log(1+k*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (3 * log(1 + k*x))^k.
Showing 1-6 of 6 results.

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