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.

Previous Showing 31-37 of 37 results.

A307126 Expansion of e.g.f. log(1 + log(1 + x*exp(x))).

Original entry on oeis.org

0, 1, 0, -2, 5, 3, -88, 362, 534, -17363, 103354, 175690, -9218328, 80446715, 46936658, -10553663682, 136009808336, -210505566343, -22766371152222, 418488315816586, -1679396876267976, -82907733267235305, 2070045795782097506, -13611715282931011890, -463120892871268874832
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 26 2019

Keywords

Crossrefs

Cf. A003713, A007550, A009306, A307125.

Programs

  • Maple
    a:=series(log(1+log(1+x*exp(x))),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + Log[1 + x Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x = 'x + O('x^30)); Vec(serlaplace(log(1 + log(1 + x*exp(x))))) \\ Michel Marcus, Mar 26 2019

A331559 E.g.f.: -log(1 + x + log(1 - x)).

Original entry on oeis.org

0, 1, 2, 9, 44, 280, 2064, 17738, 172528, 1880856, 22686960, 300193872, 4323063744, 67323469200, 1127433161568, 20205636981840, 385897245967104, 7824675262660608, 167885148101916672, 3800289634376282496, 90513807325761507840, 2262830879094971399424
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 20 2020

Keywords

Comments

Logarithmic transform of A226226.

Crossrefs

Cf. A003713, A032181, A052808, A052832, A226226, A331558.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 + x + Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

a(n) ~ (n-1)! / (1 + LambertW(-exp(-2)))^n. - Vaclav Kotesovec, Jan 26 2020

A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

Original entry on oeis.org

0, 1, 5, 29, 204, 1714, 16862, 190826, 2447512, 35136696, 558727872, 9754239648, 185546362416, 3820734689472, 84687887312688, 2010622152615504, 50908186083448320, 1369376758488222336, 38998680958184088960, 1172297572938013827456, 37092793335394301708544
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2020

Keywords

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007526, A007840, A215916, A331797.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-Log[1 - x]/((1 - x) (1 + Log[1 - x])), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[Abs[StirlingS1[n, k]] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
A007840[n_] := Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}]; a[n_] := Sum[Binomial[n, k] k! HarmonicNumber[k] A007840[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} = |Stirling1(n,k)| * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * k! * H(k) * A007840(n-k), where H(k) is the k-th harmonic number.
a(n) ~ n! / (1 - exp(-1))^(n+1). - Vaclav Kotesovec, Jan 26 2020

A336436 a(0) = 0; a(n) = ((n-1)!)^3 + (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k-1)!)^3 * k * a(k).

Original entry on oeis.org

0, 1, 5, 107, 6020, 701424, 146665984, 50005133576, 25952660212352, 19469692241358336, 20277424971134267904, 28384315863525074792448, 52002222667299924427689984, 121958564445078246232792363008, 359324017883943122680656621023232
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2020

Keywords

Crossrefs

Cf. A003713, A074708, A193420.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = ((n - 1)!)^3 + (1/n) Sum[(Binomial[n, k] (n - k - 1)!)^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[-Log[1 - Sum[x^k/k^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

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

A363115 Expansion of e.g.f. log(1 - log( sqrt(1-2*x) )).

Original entry on oeis.org

0, 1, 1, 4, 22, 168, 1616, 18800, 256432, 4012288, 70825344, 1392214272, 30157260288, 713680180224, 18319344307200, 506934586748928, 15043324048398336, 476540007615725568, 16050059458251915264, 572710950848334200832, 21582629580640554123264, 856552661738538476765184
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2023

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 22*x^4/4! + 168*x^5/5! + 1616*x^6/6! + 18800*x^7/7! + 256432*x^8/8! + 4012288*x^9/9! + ...
where
exp(A(x)) = 1 + x + 2*x^2/2 + 4*x^3/3 + 8*x^4/4 + 16*x^5/5 + ... + 2^(n-1)*x^n/n + ...

Crossrefs

Cf. A089064, A003713, A363116.

Programs

  • PARI
    {a(n) = n!*polcoeff( log((1 - log(sqrt(1-2*x +x*O(x^n))))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = (-1)^(n-1) * sum(k=1,n, 2^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = if (n<1, 0, 2^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 2^(k-1) * a(n-k)))}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = log(1 - (1/2)*log(1-2*x)).
(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 2^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
(3) a(n) = 2^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 2^(k-1) * a(n-k) for n > 0.

A363116 Expansion of e.g.f. log(1 - (1/3)*log(1-3*x)).

Original entry on oeis.org

0, 1, 2, 11, 93, 1068, 15486, 271206, 5566086, 130982328, 3476230344, 102709363392, 3343387479840, 118880973126576, 4584247231485312, 190548125567321328, 8492669888285758896, 404023626910206388224, 20434095445804056842112, 1094849162137482139541376
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2023

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 93*x^4/4! + 1068*x^5/5! + 15486*x^6/6! + 271206*x^7/7! + 5566086*x^8/8! + 130982328*x^9/9! + ...
where
exp(A(x)) = 1 + x + 3*x^2/2 + 9*x^3/3 + 27*x^4/4 + 81*x^5/5 + ... + 3^(n-1)*x^n/n + ...

Crossrefs

Cf. A089064, A003713, A363115.

Programs

  • PARI
    {a(n) = n!*polcoeff( log((1 - (1/3)*log(1-3*x +x*O(x^n) ))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = (-1)^(n-1) * sum(k=1,n, 3^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = if (n<1, 0, 3^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 3^(k-1) * a(n-k)))}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = log(1 - (1/3)*log(1-3*x)).
(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 3^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
(3) a(n) = 3^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 3^(k-1) * a(n-k) for n > 0.

A089225 Triangle T(n,k) read by rows, defined by T(n,k) = (n-k)*T(n-1,k)+Sum(k=1..n, T(n-1,k)); T(1,1) = 1, T(1,k)= 0 if k >1.

Original entry on oeis.org

1, 2, 1, 7, 4, 3, 35, 22, 17, 14, 228, 154, 122, 102, 88, 1834, 1310, 1060, 898, 782, 694, 17582, 13128, 10818, 9272, 8142, 7272, 6578, 195866, 151560, 126882, 109880, 97218, 87336, 79370, 72792, 2487832, 1981824, 1682196, 1470304, 1309776
Offset: 1

Views

Author

Philippe Deléham, Dec 10 2003

Keywords

Comments

Let M be the n X n matrix with M(i,i)=i, other entries 1. Then T(n,k) = permanent of n-1 X n-1 matrix obtained by omitting row k and column k from M.
T(n,1) = A003713(n). n-th row sum = T(n+1,n+1) = A007840(n). {1}, {2, 1}, {7, 4, 3}, {35, 22, 17, 14}, ...

Examples

			n=4: M = |1,1,1,1|1, 2,1, 1|1, 1, 3, 1|1, 1, 1, 4|
T(4, 1) = permanent of |2, 1, 1|1, 3, 1|1, 1, 4| = 26+5+4 = 35
T(4, 2) = permanent of |1, 1, 1|1, 3, 1|1, 1, 4| = 13+5+4 = 22
T(4, 3) = permanent of |1, 1, 1|1, 2, 1|1, 1, 4| = 9+5+3 = 17
T(4, 4) = permanent of |1, 1, 1|1, 2, 1|1, 1, 3| = 7+4+3 = 14
Previous Showing 31-37 of 37 results.

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