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 41-50 of 106 results. Next

A303124 Expansion of Product_{n>=1} (1 + (16*x)^n)^(1/4).

Original entry on oeis.org

1, 4, 40, 1504, 10336, 387968, 5349632, 111442944, 1100563968, 36711258112, 493805416448, 9186633203712, 134635599806464, 2648342619422720, 43443234834350080, 938422838970810368, 11378951438668791808, 224791017150689574912, 4129154423023897411584
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): A000009 (b=1), A298994 (b=2), A303074 (b=3), this sequence (b=4), A303125 (b=5).
Cf. A303131, A303153.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[-1, 16*x]/2)^(1/4), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(16*x)^k)^(1/4)))

Formula

a(n) ~ 2^(4*n - 17/8) * exp(sqrt(n/3)*Pi/2) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018

A303125 Expansion of Product_{n>=1} (1 + (25*x)^n)^(1/5).

Original entry on oeis.org

1, 5, 75, 4500, 43125, 2765000, 55871875, 1876671875, 25128437500, 1495793359375, 28953471875000, 871257974609375, 18280647500000000, 596362168603515625, 14502797130615234375, 519397373566650390625, 8604439235863037109375
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): A000009 (b=1), A298994 (b=2), A303074 (b=3), A303124 (b=4), this sequence (b=5).
Cf. A303132, A303154.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[-1, 25*x]/2)^(1/5), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(25*x)^k)^(1/5)))

Formula

a(n) ~ 5^(2*n - 1/4) * exp(Pi*sqrt(n/15)) / (2^(8/5) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018

A303130 Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).

Original entry on oeis.org

1, -3, -9, -288, 459, -19278, -1539, -1265301, 10734525, -147277926, 520204923, -7511358663, 88687160577, -668191863951, 5357547144702, -87542760890124, 967961569696722, -5115624735401361, 46065749188891275, -430898393089547667, 6203508335817169257
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): A081362 (b=1), A298993 (b=2), this sequence (b=3), A303131 (b=4), A303132 (b=5).
Cf. A303074.

Programs

  • Mathematica
    CoefficientList[Series[(2/QPochhammer[-1, 9*x])^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, (1 + (9*x)^k)^(-1/3))) \\ Altug Alkan, Apr 20 2018

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/18)) * 3^(2*n - 1/2) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A337299 Expansion of Product_{k>0} (1 - 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -2, -4, 0, -8, 16, 0, 64, 64, 384, 0, 1536, 1024, 3072, 2048, 16384, -8192, 49152, -32768, 32768, -65536, 262144, -1835008, 524288, -3145728, -6291456, -18874368, -4194304, -117440512, -16777216, -301989888, -469762048, -671088640, -805306368, -6710886400, 536870912
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2020

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = 2^(n-1).

Links

Crossrefs

Convolution inverse of A075900.
Cf. A266964, A304961.

Programs

  • Mathematica
    m = 37; CoefficientList[Series[Product[1 - 2^(k - 1)*x^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 22 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(prod(k=1, N, 1-2^(k-1)*x^k))

A022694 Expansion of Product_{m>=1} (1 + m*q^m)^-2.

Original entry on oeis.org

1, -2, -1, -2, 9, -2, 10, -16, 38, -98, 53, -116, 340, -434, 463, -990, 2378, -2792, 3660, -7058, 11454, -18900, 24104, -36206, 81623, -119400, 128194, -248062, 447066, -576154, 880401, -1415926, 2297516, -3724290, 4854450, -7299306, 13411402, -19129752, 25135890, -42841396, 71321016
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 2, g(n) = -n. - Seiichi Manyama, Dec 30 2017

Links

Crossrefs

Column k=2 of A297325.

Programs

  • Magma
    Coefficients(&*[1/(1+m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[1/(1+k*q^k)^2, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    apply(x->round(x), Vec(prodinf(m=1, 1/(1+m*q^m)^2+O(q^50)))) \\ Michel Marcus, Dec 30 2017
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n)^2)) \\ G. C. Greubel, Feb 25 2018

Formula

G.f.: exp(-2*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A066816 Expansion of Product_{k>=1} (1 + A001055(k)*x^k).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 25, 36, 46, 58, 78, 95, 120, 160, 198, 249, 318, 392, 485, 608, 745, 914, 1140, 1390, 1692, 2092, 2528, 3032, 3709, 4468, 5364, 6494, 7770, 9279, 11161, 13347, 15824, 18920, 22465, 26539, 31607, 37345, 43994, 52016
Offset: 0

Views

Author

Vladeta Jovovic, Jan 20 2002

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A001055(n). - Seiichi Manyama, Nov 14 2018

Links

Crossrefs

Cf. A001055, A066739, A066806.

Formula

a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), n > 0, a(0)=1, b(k) = Sum_{d|k} (-1)^(k/d+1)*d*(A001055(d))^(k/d).

A294580 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^j.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, 0, 1, -1, -16, -19, -3, 4, 1, -1, -32, -65, -21, 23, 4, 1, -1, -64, -211, -111, 139, 44, 7, 1, -1, -128, -665, -525, 863, 448, 104, 3, 1, -1, -256, -2059, -2343, 5419, 4316, 1414, 70, -2, 1, -1, -512, -6305, -10101, 34103, 40024, 18164, 1206, -93, -9
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,    1,    1, ...
   -1, -1,  -1,   -1,   -1, ...
   -2, -4,  -8,  -16,  -32, ...
   -1, -5, -19,  -65, -211, ...
    0, -3, -21, -111, -525, ...

Links

Crossrefs

Columns k=0..2 give A073592, A266964, A294581.
Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079(n+1), (-1)*A001047(n+1).
Cf. A292166, A294582.

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.

A294583 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^(j^k).

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -5, 0, 1, -1, -64, -65, -3, 1, 1, -1, -256, -665, -79, 23, 0, 1, -1, -1024, -6305, -1575, 831, 44, 1, 1, -1, -4096, -58025, -28255, 33335, 4789, 104, 0, 1, -1, -16384, -527345, -481623, 1323807, 411664, 15099, 70, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,     1,      1, ...
   -1, -1,  -1,    -1,     -1, ...
   -1, -4, -16,   -64,   -256, ...
    0, -5, -65,  -665,  -6305, ...
    0, -3, -79, -1575, -28255, ...

Links

Crossrefs

Columns k=0..2 give A010815, A266964, A294584.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294585.

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+k*j/d)) * A(n-j,k) for n > 0.

A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, -1, 1, -1, -16, -19, -3, 5, 1, -1, -32, -65, -13, 23, 1, 1, -1, -64, -211, -63, 131, 44, 13, 1, -1, -128, -665, -301, 815, 497, 104, 4, 1, -1, -256, -2059, -1383, 5195, 4840, 1149, 70, 0, 1, -1, -512, -6305, -6133, 33143, 45021, 13752, 662, -93, 2
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,   1,    1, ...
   -1, -1,  -1,  -1,   -1, ...
   -2, -4,  -8, -16,  -32, ...
   -1, -5, -19, -65, -211, ...
   -1, -3, -13, -63, -301, ...

Links

Crossrefs

Columns k=0..2 give A022661, A266964, A294588.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283272.

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+j/d)) * A(n-j,k) for n > 0.

A294957 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^(k-1)).

Original entry on oeis.org

1, 1, 5, 32, 300, 3533, 51650, 894929, 17981196, 410826036, 10518152538, 298209605418, 9273131902539, 313758357802886, 11474239675400172, 450962279143408815, 18954601400362304902, 848385358833157331498, 40285279861744621069122
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2017

Keywords

Comments

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

Links

Crossrefs

Cf. A294956.

Programs

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

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294956(k)*a(n-k) for n > 0.
Previous Showing 41-50 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.