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 21-28 of 28 results.

A331558 E.g.f.: log(1 - x - log(1 - x)).

Original entry on oeis.org

0, 1, 2, 3, 4, 20, 216, 1862, 13840, 104904, 949200, 10517232, 130307904, 1694503824, 23063845728, 335427395760, 5269616092416, 88835797577472, 1587229554415104, 29838489410093184, 589394278657267200, 12235109311726689024, 266570965998899071488
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 20 2020

Keywords

Comments

Logarithmic transform applied twice to A000166.

Crossrefs

Cf. A000166, A052808, A089064, A331559.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 - x - Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

A341598 a(n) = Sum_{k=n..2*n} |Stirling1(2*n, k)| * Stirling1(k, n).

Original entry on oeis.org

1, 0, 4, 15, 728, 9660, 454333, 11921910, 620800752, 25052417676, 1495629968820, 81260657073596, 5594820193907943, 379090865741895580, 29938401724408721880, 2414113646907092768775, 216602054576835471646080, 20165486015516015341186800, 2034029167741961519973600460
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 15 2021

Keywords

Links

Crossrefs

Cf. A008275, A079642, A089064, A321712, A341575, A341589.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[2 n, k]] StirlingS1[k, n], {k, n, 2 n}], {n, 0, 18}]
Table[((2 n)!/n!) SeriesCoefficient[Log[1 - Log[1 - x]]^n, {x, 0, 2 n}], {n, 0, 18}]
  • PARI
    a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1))*stirling(k, n, 1)); \\ Michel Marcus, Feb 16 2021

Formula

a(n) = ((2*n)!/n!) * [x^(2*n)] log(1 - log(1 - x))^n.
a(n) ~ c * d^n * (n-1)!, where d = 5.87606029984908... and c = 0.08380514489... - Vaclav Kotesovec, Feb 17 2021

A354416 Expansion of e.g.f. (1 - log(1-x))^x.

Original entry on oeis.org

1, 0, 2, 0, 16, 5, 288, 392, 9840, 33462, 582910, 3652044, 55557192, 524095728, 7910319116, 98390834310, 1573086910848, 23774700449584, 414180226506456, 7249907657342184, 138771378745878680, 2735366111451910944, 57469663931297252976, 1253755421949789141624
Offset: 0

Views

Author

Seiichi Manyama, May 26 2022

Keywords

Crossrefs

Cf. A089064, A351739, A354083.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-log(1-x))^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^j*j*sum(k=1, j-1, (k-1)!*stirling(j-1, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} k * A089064(k-1) * binomial(n-1,k-1) * a(n-k).
a(n) ~ (n-1)!. - Vaclav Kotesovec, Jun 08 2022

A360067 a(n) = det(M) where M is an n X n matrix with M[i,j] = i^j*(i-j).

Original entry on oeis.org

1, 0, 2, 12, 2304, 898560, 4827340800, 143219736576000, 49230909076930560000, 149334225705682285363200000, 5482643392499167214520238080000000, 2322479608280149573505226859610112000000000, 13283541711093841017468807905468592685056000000000000
Offset: 0

Views

Author

José María Grau Ribas, Jan 24 2023

Keywords

Crossrefs

Cf. A060238, A174890, A176005, A176001, A000178, A089064, A152653.

Programs

  • Maple
    a:= n-> LinearAlgebra[Determinant](Matrix(n, (i,j) -> i^j*(i-j))):
seq(a(n), n=0..12);  # Alois P. Heinz, Jan 25 2023
  • Mathematica
    a[n_] := Det@Table[i^j (i - j), {i, n}, {j, n}]; Table[a[n], {n, 1, 15}]
  • PARI
    a(n) = matdet(matrix(n, n, i, j, i^j*(i-j))); \\ Michel Marcus, Jan 24 2023
  • Python
    from sympy import Matrix
def A360067(n): return Matrix(n,n,[i**j*(i-j) for i in range(1,n+1) for j in range(1,n+1)]).det() # Chai Wah Wu, Jan 27 2023

Formula

For n>=1, a(n) = A000178(n-1) * A089064(n). - Vaclav Kotesovec, Apr 19 2024

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.

A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

Original entry on oeis.org

0, 0, 2, 3, -4, -30, 54, 1260, 3856, -36288, -279000, 2970000, 56725008, 109343520, -5495740992, -26086263840, 1293641890560, 21771049466880, -45508965806592, -4589738336217600, 10493846174810880, 2423866077943511040, 34328754265480012800, -358930542362135546880
Offset: 0

Views

Author

Seiichi Manyama, Jan 21 2025

Keywords

Crossrefs

Cf. A052804, A052858.
Cf. A089064, A380339.
Cf. A001048, A375684.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(log(1-x*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=1, n\2, (-1)^(k-1)*(k-1)!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * (k-1)! * |Stirling1(n-k,k)|/(n-k)!.
a(0) = a(1) = 0; a(n) = n * (n-2)! - Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k).

A380339 Expansion of e.g.f. log(1 - x^2/2 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 0, -126, -1260, 3240, 108360, 1635480, 15075720, 119957760, 705729024, 6324040800, 130989549600, 3572031415680, 78736127656320, 1502102645890560, 25514633892182400, 423898384988494080, 7590291773745542400, 162254912688831916800, 4023271392778314673920
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2025

Keywords

Crossrefs

Cf. A089064, A380338.
Cf. A368165, A368173.

Programs

  • PARI
    a(n) = n!*sum(k=1, n\3, (-1)^(k-1)*(k-1)!*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/3)} (-1)^(k-1) * (k-1)! * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
a(0) = a(1) = a(2) = 0; a(n) = n!/(2*(n-2)) - Sum_{k=3..n-1} k!/(2*(k-2)) * binomial(n-1,k) * a(n-k).
Previous Showing 21-28 of 28 results.

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