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 1-7 of 7 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...

A301581 G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 2, 16, 232, 4748, 125440, 4058312, 155336672, 6864980968, 343995674280, 19270975801600, 1193481831243584, 80966964261458368, 5971270693661978816, 475655179279901897536, 40699219246551726635840, 3722813577249648564213392, 362519587815189751405383520, 37442485808471509306691295808, 4088344078912544484116541775616, 470550859964811044524886252649760
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 232*x^3 + 4748*x^4 + 125440*x^5 + 4058312*x^6 + 155336672*x^7 + 6864980968*x^8 + 343995674280*x^9 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 4*((1+x)^2-1)^2 + 8*((1+x)^3-1)^3 + 16*((1+x)^4-1)^4 + 32*((1+x)^5-1)^5 + 64*((1+x)^6-1)^6 + 128*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/3 + 2*(1+x)/(1 + 2*(1+x))^2 + 4*(1+x)^4/(1 + 2*(1+x)^2)^3 + 8*(1+x)^9/(1 + 2*(1+x)^3)^4 + 16*(1+x)^16/(1 + 2*(1+x)^4)^5 + 32*(1+x)^25/(1 + 2*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A122400, A195005, A301582, A301583, A301463.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[2^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, 2^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) /(1 + 2*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 2*exp(1/r)) * r^2 = 5.90397654567002180048849478643459748789951071189960380570421209622277299181878..., 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 = 0.3270084259351451234078681822585067892123772641092056... - Vaclav Kotesovec, Aug 09 2018, updated Oct 08 2020

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

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

A350719 a(n) = Sum_{k=0..n} k! * 2^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 2, 30, 1108, 76372, 8463328, 1375868768, 308440047648, 91189383264864, 34376022491122368, 16093445542120281792, 9160424435706947112576, 6230035512106223752576896, 4989402076922846372194268160, 4647526704475074504983564884992
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A320083, A350720.
Cf. A088501, A195005, A350721.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 2^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*2^k*k^n*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*log(1+k*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (2 * log(1 + k*x))^k.

A338044 E.g.f.: Sum_{j>=0} 2^j * (1 - exp(-j*x))^j.

Original entry on oeis.org

1, 2, 30, 1106, 75870, 8355602, 1349011230, 300225115346, 88096432294110, 32956583516814482, 15309575613991708830, 8646194423981547656786, 5834064910665307876000350, 4635347672272868599469126162, 4283458291212292843946379302430
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 08 2020

Keywords

Links

Crossrefs

Cf. A195005, A220181.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[2^j*(1 - Exp[-j*x])^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[(-1)^(n-j) * 2^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*2^k*k^n*k!*stirling(n, k, 2)); \\ Seiichi Manyama, Jan 31 2022

Formula

a(n) = Sum_{j=0..n} (-1)^(n-j) * 2^j * j^n * j! * Stirling2(n,j).
a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 4.888902442941545347850916031937657541653741222401134656609725875258275714... and c = 0.4779849579705948535026794982366398948961135521828033628215401277586...
Showing 1-7 of 7 results.

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