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.

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

Original entry on oeis.org

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846
Offset: 0

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

Keywords

Links

Crossrefs

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015
G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

Extensions

More terms from Vaclav Kotesovec, Aug 25 2015

A261567 Expansion of Product_{k>=1} (1/(1 + 3*x^k))^k.

Original entry on oeis.org

1, -3, 3, -18, 69, -168, 504, -1578, 4800, -14310, 42396, -128049, 385839, -1154271, 3458847, -10386477, 31173873, -93490386, 280426833, -841384614, 2524300014, -7572585150, 22717270491, -68152872885, 204460229394, -613377236379, 1840126774737, -5520391488054
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

Comments

In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

Links

Crossrefs

Cf. A255528, A261566, A261582.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1/(1 + 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1 - 1/(-3)^j)^(j+1) = 0.72392917591300902192520561680114697538581509655711959502191898288595312452...

A343465 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-3)^d.

Original entry on oeis.org

3, -3, 11, -21, 51, -119, 315, -831, 2195, -5883, 16107, -44357, 122643, -341487, 956635, -2690841, 7596483, -21522347, 61171659, -174342165, 498112275, -1426403751, 4093181691, -11767920107, 33891544419, -97764009003, 282429537947, -817028472645, 2366564736723, -6863037262207
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 16 2021

Keywords

Crossrefs

Cf. A000010, A001867, A032308, A038064, A074763, A261582, A343466, A343467.

Programs

  • Mathematica
    Table[-(1/n) Sum[EulerPhi[n/d] (-3)^d, {d, Divisors[n]}], {n, 1, 30}]
nmax = 30; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 3 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} phi(k) * log(1 + 3*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-3)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A032308.
Product_{n>=1} (1 - x^n)^a(n) = g.f. for A261582.

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2022

Keywords

Crossrefs

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

Formula

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts.

A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, -1, 0, 1, -4, 6, -6, 1, 0, 1, -5, 12, -21, 14, -1, 0, 1, -6, 20, -52, 69, -26, 1, 0, 1, -7, 30, -105, 220, -201, 50, -1, 0, 1, -8, 42, -186, 545, -868, 591, -102, 2, 0, 1, -9, 56, -301, 1146, -2705, 3436, -1785, 214, -2, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1, ...
   0, -1, -2,  -3,  -4, ...
   0,  0,  2,   6,  12, ...
   0, -1, -6, -21, -52, ...
   0,  1, 14,  69, 220, ...

Links

Crossrefs

Columns k=0..3 give A000007, A081362, A071109, A261582.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292134.
Cf. A246935, A286957.

A351024 Dirichlet g.f.: Product_{k>=2} 1 / (1 + 3 * k^(-s)).

Original entry on oeis.org

1, -3, -3, 6, -3, 6, -3, -21, 6, 6, -3, -12, -3, 6, 6, 69, -3, -12, -3, -12, 6, 6, -3, 51, 6, 6, -21, -12, -3, -3, -3, -201, 6, 6, 6, 33, -3, 6, 6, 51, -3, -3, -3, -12, -12, 6, -3, -156, 6, -12, 6, -12, -3, 51, 6, 51, 6, 6, -3, 15, -3, 6, -12, 591, 6, -3, -3, -12, 6, -3
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 29 2022

Keywords

Crossrefs

Cf. A261582, A316441, A339718, A349922, A351023.

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -3, -6, -18, -27, -108, -81, -486, 0, -1458, 8748, -6561, 118098, 118098, 1003833, 1417176, 11691702, 9565938, 105225318, 114791256, 746143164, 1076168025, 7231849128, 2324522934, 58113073350, 45328197213, 334731302496, 146444944842, 3263630199336, -3012581722464
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2022

Keywords

Crossrefs

Cf. A003056, A032308, A242587, A261582, A292128, A300579, A337299, A344062, A352762.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 30}]

Formula

a(n) = Sum_{k=0..A003056(n)} (-1)^k * q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
Showing 1-7 of 7 results.

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