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 1-9 of 9 results.

A193436 exp( Sum_{n>=1} x^n/n^3 ) = Sum_{n>=0} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 1, 5, 71, 2276, 144724, 16688884, 3249507820, 1005334796864, 468967172341824, 315409074574480704, 294510517409159769024, 369877735410388416241920, 608401340784471133062837504, 1281569707473914769353921666304, 3391681347749396029674738480747264
Offset: 0

Views

Author

Paul D. Hanna, Jul 25 2011

Keywords

Comments

Sum_{n>=0} a(n)/n!^3 = exp(zeta(3)) = 3.326953110002499790...

Examples

			A(x) = 1 + x + 5*x^2/2!^3 + 71*x^3/3!^3 + 2276*x^4/4!^3 +...
where
log(A(x)) = x + x^2/8 + x^3/27 + x^4/64 + x^5/125 + x^6/216 +...

Crossrefs

Cf. A074707, A217145, A193435.

Programs

  • PARI
    {a(n)=n!^3*polcoeff(exp(sum(m=1,n,x^m/m^3)+x*O(x^n)),n)}

Formula

a(0) = 1; a(n) = (n-1)! * (n!)^2 * Sum_{k=0..n-1} a(k) / ((k!)^3 * (n-k)^2). - Ilya Gutkovskiy, Jul 18 2020

A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

Original entry on oeis.org

1, 1, 5, 58, 1208, 39476, 1861372, 119587224, 10040970816, 1067383279872, 140110136642304, 22256626639796352, 4207858001708629248, 933704296260740939520, 240293228328619963492608, 70992050129486593239246336, 23863916105454465092261412864
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2020

Keywords

Links

Crossrefs

Cf. A007840, A074707, A102221, A336243, A336259, A336260, A336261.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^2, i=1..n))
    end:
a:= n-> n!^2*b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Jan 04 2024
  • Mathematica
    a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - polylog(2,x)).
a(n) ~ (n!)^2 / (-log(1-r) * r^n), where r = 0.76154294453204558806805187241... is the root of the equation polylog(2,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A217145 exp( Sum_{n>=1} x^n/n^4 ) = Sum_{n>=0} a(n)*x^n/n!^4.

Original entry on oeis.org

1, 1, 9, 313, 30232, 6874776, 3355094696, 3302015131304, 6189229701416448, 20757720442141804032, 116803259505967824465408, 1039413737809909553149398528, 13914325979093456341597993070592, 268988472559744572003351007811825664
Offset: 0

Views

Author

Paul D. Hanna, Oct 18 2012

Keywords

Comments

Sum_{n>=0} a(n)/n!^4 = exp(Pi^4/90) = 2.951528682853355...

Examples

			A(x) = 1 + x + 9*x^2/2!^4 + 313*x^3/3!^4 + 30232*x^4/4!^4 + 6874776*x^5/5!^4 +...
where
log(A(x)) = x + x^2/2^4 + x^3/3^4 + x^4/4^4 + x^5/5^3 + x^6/6^4 +...

Crossrefs

Cf. A074707, A193436.

Programs

  • PARI
    {a(n)=n!^4*polcoeff(exp(sum(m=1, n, x^m/m^4)+x*O(x^n)), n)}
for(n=0,20,print1(a(n),", "))

Formula

a(0) = 1; a(n) = (n-1)! * (n!)^3 * Sum_{k=0..n-1} a(k) / ((k!)^4 * (n-k)^3). - Ilya Gutkovskiy, Jul 18 2020

A346291 a(0) = 1; a(n) = (1/n) * Sum_{k=2..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

Original entry on oeis.org

1, 0, 1, 4, 54, 976, 27050, 1037016, 53040344, 3494603904, 288738690552, 29267185135200, 3573720291756912, 517691602686711168, 87813773085480166608, 17246816939881695262656, 3883816372280829757142400, 994217196872849143760818176, 287129874355801742457562921344
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 13 2021

Keywords

Crossrefs

Cf. A000166, A074707, A346292.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 2, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x ).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=2} x^n / n^2 ).

A336289 a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 5, 55, 1054, 31046, 1299386, 73211510, 5338080280, 488727800664, 54865512897432, 7408400404206792, 1184230737883333680, 221121985937352261360, 47683177920267470877648, 11758982455716373002624816, 3287966057434181416523799936
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

Crossrefs

Cf. A001008, A002805, A074707, A087761, A235385, A235685, A336290.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] (k - 1)! HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; CoefficientList[Series[Exp[Log[1 - x]^2/2 + PolyLog[2, x]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} H(n) * x^n / n).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(log(1 - x)^2 / 2 + polylog(2,x)).

A346292 a(0) = 1; a(n) = (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

Original entry on oeis.org

1, 0, 0, 4, 36, 576, 17600, 694800, 35802144, 2391438336, 200018045952, 20476348214400, 2521840589347200, 368057828019898368, 62841061478699292672, 12413136137144581203456, 2809529229255558769612800, 722458985698006017844838400, 209487621780682072569567903744
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 13 2021

Keywords

Crossrefs

Cf. A038205, A074707, A346291.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 3, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x - x^2 / 4 ).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=3} x^n / n^2 ).

A336441 a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / k^n).

Original entry on oeis.org

1, 1, 3, 71, 30232, 435772624, 357189846148256, 25740403176657987904960, 234446578865185870182814945640448, 363178754511398964104990417951192651478859776, 122088173887703514886799765831338556792096849201928981184512
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2020

Keywords

Crossrefs

Cf. A000142, A074707, A193436, A217145, A275044.

Programs

  • Mathematica
    Table[(n!)^n SeriesCoefficient[Exp[Sum[x^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 1, (1/n) Sum[(Binomial[n, j] (n - j - 1)!)^k (n - j) b[j, k], {j, 0, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

Formula

From Vaclav Kotesovec, Oct 28 2024: (Start)
a(n) ~ (n!)^(n-1).
a(n) ~ (2*Pi)^((n-1)/2) * n^(n^2 - n/2 - 1/2) / exp(n^2 - n - 1/12). (End)

A346371 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / (2*n+1)^2 ).

Original entry on oeis.org

1, 1, 2, 10, 88, 1496, 34256, 1305872, 57804160, 3960382848, 288097804032, 31177032137472, 3374496463248384, 530644850402565120, 79955455534325999616, 17241179374803330287616, 3448609425518084068048896, 977269122457749276877750272, 250420488297020919542581493760
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 14 2021

Keywords

Crossrefs

Cf. A000246, A074707.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[Sum[x^(2 k + 1)/(2 k + 1)^2, {k, 0, Infinity}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, 2 k + 1] (2 k + 1)!)^2 a[n - 2 k - 1]/(2 k + 1), {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} (binomial(n,2*k+1) * (2*k+1)!)^2 * a(n-2*k-1) / (2*k+1).

A346372 a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

Original entry on oeis.org

1, 1, 2, 10, 124, 2396, 64856, 2452472, 124483360, 8146185504, 668645524032, 67374446014272, 8183368905811584, 1179807474740449920, 199266648878034317568, 38984601149045449948416, 8748103140554862876727296, 2232274640259371687436982272, 642805438643602793466093711360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 14 2021

Keywords

Crossrefs

Cf. A000266, A074707, A346291.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 3, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x^2 / 4 ).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / n^2 ).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.