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 21-30 of 106 results. Next

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

Original entry on oeis.org

1, -1, -8, -73, -927, -13969, -254580, -5288596, -124795126, -3272571133, -94692028369, -2991756529687, -102571647087930, -3791499758414848, -150359326161180392, -6367668575791613601, -286854342016830115157, -13697147209040205869792
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = n^n.

Links

Crossrefs

Column k=1 of A294808.
Cf. A294810.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k^k*x^k)^k))

Formula

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294810(k)*a(n-k) for n > 0.

A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -2, -4, 0, 3, 17, 24, 40, 9, -24, -149, -250, -435, -395, -281, 514, 1528, 3542, 5127, 6920, 5416, 1368, -11136, -28533, -57051, -82846, -107315, -95655, -43646, 107826, 345877, 727771, 1150968, 1601729, 1766547, 1495154, 183944, -2339567, -6770991, -12701854
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 09 2017

Keywords

Comments

Convolution inverse of A028377.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(n+1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Links

Crossrefs

Cf. A000217, A000294, A028377, A255528, A284896, A292386.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^A000217(k).
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(d+1)*(-1)^(n/d). - Seiichi Manyama, Nov 14 2017

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024
Offset: 0

Views

Author

William J. Keith, Jan 18 2018

Keywords

Comments

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A000108, A271235, A298994.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Programs

  • Mathematica
    Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]
  • PARI
    q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).

A303135 Expansion of Product_{n>=1} (1 - (16*x)^n)^(-1/4).

Original entry on oeis.org

1, 4, 104, 1760, 39520, 590720, 14285056, 205151232, 4596467200, 75375073280, 1504196046848, 23673049726976, 525315968712704, 7912159583600640, 158055039529779200, 2726833423421800448, 51889395654107463680, 840470097284214292480, 16765991910040314839040
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/4, g(n) = 16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), A271236 (b=3), this sequence (b=4), A303136 (b=5).
Cf. A303131, A303153.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[16*x]^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)

Formula

a(n) ~ exp(sqrt(n/6)*Pi) * 2^(4*n - 33/16) / (3^(5/16) * n^(13/16)). - Vaclav Kotesovec, Apr 19 2018

A303136 Expansion of Product_{n>=1} (1 - (25*x)^n)^(-1/5).

Original entry on oeis.org

1, 5, 200, 5125, 177500, 3952500, 150715625, 3185187500, 112844843750, 2783033593750, 86330708203125, 2019237027343750, 72195817812500000, 1591910699609375000, 50158322275878906250, 1322261581989501953125, 39183430287559814453125, 946961406814801025390625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), A271236 (b=3), A303135 (b=4), this sequence (b=5).
Cf. A303132, A303154.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[25*x]^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)
CoefficientList[Series[Product[(1-(25x)^n)^(-1/5),{n,20}],{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/15)) * 5^(2*n - 3/10) / (2^(7/5) * 3^(3/10) * n^(4/5)). - Vaclav Kotesovec, Apr 19 2018

A303153 Expansion of Product_{n>=1} (1 - (16*x)^n)^(1/4).

Original entry on oeis.org

1, -4, -88, -992, -19360, -97152, -4296448, 4539392, -568015360, -127621120, -39357927424, 2424998313984, -38804685471744, 799759166930944, 4879962868940800, 41563181340426240, 585185165832486912, 55834295603426754560, -75535223925056208896
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/4, g(n) = 16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), this sequence (b=4), A303154 (b=5).
Cf. A303124, A303135.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(16*x)^k)^(1/4)))

A303154 Expansion of Product_{n>=1} (1 - (25*x)^n)^(1/5).

Original entry on oeis.org

1, -5, -175, -3250, -100625, -1015000, -58034375, -154171875, -22257500000, -154144921875, -6824828906250, 175448177734375, -8774446542968750, 164769756689453125, 756859169189453125, 9661555852294921875, -16148589271240234375, 81663068586871337890625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), A303153 (b=4), this sequence (b=5).
Cf. A303125, A303136.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(25*x)^k)^(1/5)))

A303131 Expansion of Product_{n>=1} (1 + (16*x)^n)^(-1/4).

Original entry on oeis.org

1, -4, -24, -1248, 1632, -267136, -669440, -56925184, 597165568, -19934894080, 61831327744, -3209599664128, 47593545383936, -840449808072704, 8113679782510592, -350055154021040128, 5703847053344768000, -57129722970675609600, 704939718429511778304
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/4, g(n) = -16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), this sequence (b=4), A303132 (b=5).
Cf. A303124, A303135.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 16*x])^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/24)) * 2^(4*n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A303132 Expansion of Product_{n>=1} (1 + (25*x)^n)^(-1/5).

Original entry on oeis.org

1, -5, -50, -3875, 2500, -2046250, -12409375, -1087687500, 13232343750, -907225000000, 1545669140625, -362705679687500, 6007095839843750, -224713698632812500, 2118331116210937500, -226812683210205078125, 4765872641563720703125
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/5, g(n) = -25^n.
In general, for h>=1, if g.f. = Product_{k>=1} (1 + (h^2*x)^k)^(-1/h), then a(n) ~ (-1)^n * exp(Pi*sqrt(n/(6*h))) * h^(2*n) / (2^(7/4) * 3^(1/4) * h^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

Links

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), A303131 (b=4), this sequence (b=5).
Cf. A303125, A303136.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 25*x])^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/30)) * 5^(2*n - 1/4) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A303152 Expansion of Product_{n>=1} (1 - (9*x)^n)^(1/3).

Original entry on oeis.org

1, -3, -36, -207, -2214, -2754, -138591, 547722, -3730293, 30138075, 133709535, 7735237479, -35284817430, 702841889322, 3056530613769, 9493893988155, 112554319443867, 3822223052352735, -3940051663965051, 250298859930263181, -551418001934739786, 1061747224529191191
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/3, g(n) = 9^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), this sequence (b=3), A303153 (b=4), A303154 (b=5).
Cf. A271236, A303074.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(9*x)^k)^(1/3)))
Previous Showing 21-30 of 106 results. Next

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

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