A383818 -id:A383818 - cp's OEIS frontend

A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

3, 7, 0, 4, 2, 1, 1, 7, 5, 6, 3, 3, 9, 2, 6, 7, 9, 8, 4, 9, 5, 7, 4, 3, 1, 8, 9, 4, 1, 1, 2, 6, 8, 1, 0, 0, 9, 7, 8, 1, 2, 8, 5, 9, 6, 7, 8, 4, 6, 0, 5, 3, 3, 4, 8, 1, 5, 3, 8, 8, 6, 0, 2, 7, 8, 1, 5, 4, 3, 8, 6, 7, 8, 3, 1, 5, 7, 3, 5, 1, 5, 6, 5, 6, 0, 1, 0

Offset: 0

Artur Jasinski, May 11 2025

The real root of the cubic polynomial 729*x^3 - 810*x^2 - 429*x + 233 matches this constant to 20 decimal places.

			0.3704211756339267984957431894112681...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2004(2020), page 9, eq. (3.1h), put q = 1/3 and alpha = 3. - _Amiram Eldar_, May 11 2025

Cf. A007404, A055777, A383818, A383819, A383820.

sum(k=1,logint(2^getlocalbitprec(),3)+1,moebius(3*k)/(3.^k - 1),0.) \\ Bill Allombert

Equals Sum_{k>=0} 1/3^(3^k) = Sum_{k>=0} 1/A055777(k). - Amiram Eldar, May 11 2025

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 12, 18, 2, 0, 1, 20, 72, 42, 2, 0, 1, 30, 200, 312, 90, 2, 0, 1, 42, 450, 1400, 1152, 186, 2, 0, 1, 56, 882, 4650, 8000, 3912, 378, 2, 0, 1, 72, 1568, 12642, 38250, 40520, 12672, 762, 2, 0, 1, 90, 2592, 29792, 142002, 271770, 190400, 39912, 1530, 2, 0

Offset: 0

Seiichi Manyama, May 14 2025

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  0, 2,   6,   12,    20,     30, ...
  0, 2,  18,   72,   200,    450, ...
  0, 2,  42,  312,  1400,   4650, ...
  0, 2,  90, 1152,  8000,  38250, ...
  0, 2, 186, 3912, 40520, 271770, ...

Columns k=0..4 give A000007, A040000, A068293(n+1), A383910, A383911.
Main diagonal gives A350366.
A(n,n-1) gives A383767.
Cf. A383818, A383843.

a(n, k) = sum(j=0, k, abs(stirling(k+1, j+1, 1))*stirling(j+n, k, 2));

A(n,k) = Sum_{j=0..k} |Stirling1(k+1,j+1)| * Stirling2(j+n,k).

A383912 Expansion of (1+x) * (1+2x)/((1-x) (1-2x) (1-3*x)).

Original entry on oeis.org

1, 9, 45, 177, 621, 2049, 6525, 20337, 62541, 190689, 578205, 1746897, 5265261, 15844929, 47633085, 143095857, 429680781, 1289828769, 3871059165, 11616323217, 34855261101, 104578366209, 313760264445, 941331124977, 2824094038221, 8472483441249, 25417852976925

Offset: 0

Seiichi Manyama, May 15 2025

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Column k=3 of A383818.

Cf. A383910.

PARI
```
a(n) = 10*3^n-3*2^(n+2)+3;
```

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
a(n) = Sum_{k=0..3} |Stirling1(3,k)| * Stirling2(k+n,3).
a(n) = 10*3^n - 3*2^(n+2) + 3.

A383913 Expansion of (1+x) * (1+2x) (1+3x)/((1-x) (1-2x) (1-3x) (1-4*x)).

Original entry on oeis.org

1, 16, 136, 856, 4576, 22216, 101536, 446056, 1907776, 8009416, 33187936, 136233256, 555438976, 2253396616, 9108754336, 36721012456, 147743018176, 593550943816, 2381944320736, 9551006783656, 38273731365376, 153304069611016, 613843773807136, 2457257707146856

Offset: 0

Seiichi Manyama, May 15 2025

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Column k=4 of A383818.

Cf. A383911.

PARI
```
a(n) = 35*4^n-20*3^(n+1)+15*2^(n+1)-4;
```

a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).
a(n) = Sum_{k=0..4} |Stirling1(4,k)| * Stirling2(k+n,4).
a(n) = 35*4^n - 20*3^(n+1) + 15*2^(n+1) - 4.

cp's OEIS Frontend

A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

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A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

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A383912 Expansion of (1+x) * (1+2x)/((1-x) (1-2x) (1-3*x)).

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PARI

Formula

A383913 Expansion of (1+x) * (1+2x) (1+3x)/((1-x) (1-2x) (1-3x) (1-4*x)).

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cp's OEIS Frontend

A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + j*x)/(1 - j*x).

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PARI

Formula

A383912 Expansion of (1+x) * (1+2*x)/((1-x) * (1-2*x) * (1-3*x)).

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A383913 Expansion of (1+x) * (1+2*x) * (1+3*x)/((1-x) * (1-2*x) * (1-3*x) * (1-4*x)).

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Author

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PARI

Formula

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

A383912 Expansion of (1+x) * (1+2x)/((1-x) (1-2x) (1-3*x)).

A383913 Expansion of (1+x) * (1+2x) (1+3x)/((1-x) (1-2x) (1-3x) (1-4*x)).