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 1-10 of 10 results.

A160461 Coefficients in the expansion of C/B^2, in Watson's notation of page 106.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 63, 105, 175, 280, 444, 685, 1050, 1575, 2345, 3439, 5005, 7195, 10275, 14525, 20405, 28428, 39375, 54150, 74080, 100715, 136265, 183365, 245645, 327485, 434810, 574790, 756965, 992950, 1297940, 1690500, 2194642, 2839695, 3663225, 4711160
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^3+2*x^27+5*x^51+10*x^75+20*x^99+35*x^123+63*x^147+...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): this sequence (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).
Cf. A000041, A015128, A278690, A298311.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(3)*exp(sqrt(6*n/5)*Pi)/(20*n). - Vaclav Kotesovec, Nov 26 2016
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 5. Cf. A000041 (k = 1), A015128 (k = 2), A278690 (k = 3) and A298311 (k = 4). - Peter Bala, Nov 17 2020

A160460 Coefficients in the expansion of C^6 / B^7, in Watson's notation of page 106.

Original entry on oeis.org

1, 7, 35, 140, 490, 1541, 4480, 12195, 31465, 77525, 183626, 420077, 932030, 2011905, 4237130, 8725671, 17605602, 34861815, 67848095, 129946805, 245203642, 456303872, 838178470, 1520969100, 2728472695, 4841909821, 8504898720, 14794863270, 25500965320
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^23 + 7*x^47 + 35*x^71 + 140*x^95 + 490*x^119 + 1541*x^143 + ...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), this sequence (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^6/(1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(29/15) * exp(Pi*sqrt(58*n/15)) / (500*n). - Vaclav Kotesovec, Nov 28 2016

A278555 Expansion of Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13 in powers of x.

Original entry on oeis.org

1, 13, 104, 637, 3276, 14808, 60541, 228124, 803010, 2667054, 8422715, 25446304, 73907808, 207209614, 562673618, 1484147681, 3811882087, 9553588317, 23407932874, 56161135485, 132132608899, 305240006266, 693150485885, 1548871015291, 3408852663762, 7395582677152
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 24 2016

Links

Crossrefs

Cf. A160460, A278556, A278557, A278558, A278559.

Programs

  • Mathematica
    CoefficientList[ Series[ Product[(1 - x^(5n))^12/(1 - x^n)^13, {n, 25}],
{x, 0, 25}], x] (* Robert G. Wilson v, Nov 23 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*a(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(53/15)*exp(sqrt(106*n/15)*Pi)/(62500*n). - Vaclav Kotesovec, Nov 24 2016

A278557 Expansion of Product_{n>=1} (1 - x^(5*n))^24/(1 - x^n)^25 in powers of x.

Original entry on oeis.org

1, 25, 350, 3575, 29575, 209381, 1312075, 7443825, 38854075, 188836375, 862496902, 3729343275, 15356254650, 60511763600, 229125615600, 836555203223, 2953900713000, 10113407774450, 33649438734125, 109017926343725, 344525085375315, 1063718962906450
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Links

Crossrefs

Cf. A160460, A278555, A278556, A278558, A278559.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^24/(1 - x^k)^25, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^24/(1 - x^n)^25.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*a(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(101/15) * exp(Pi*sqrt(202*n/15)) / (976562500*n). - Vaclav Kotesovec, Nov 28 2016

A278558 Expansion of Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31 in powers of x.

Original entry on oeis.org

1, 31, 527, 6448, 63240, 526443, 3852742, 25380847, 153068700, 855816380, 4479330091, 22117432019, 103672066076, 463698703204, 1987628351600, 8195086588810, 32603090921532, 125497791966435, 468512597653134, 1699911932127300, 6005651320362628, 20693956328627358
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 28 2016

Links

Crossrefs

Cf. A160460, A278555, A278556, A278557, A278559.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^30/(1 - x^k)^31, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*a(n-4) for n >= 4.
a(n) ~ exp(Pi*5*sqrt(2*n/3)) / (24414062500*sqrt(3)*n). - Vaclav Kotesovec, Nov 28 2016

A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.

Original entry on oeis.org

1, 3, 9, 22, 51, 106, 215, 411, 766, 1377, 2423, 4154, 7001, 11567, 18834, 30195, 47809, 74735, 115585, 176847, 268064, 402598, 599695, 886116, 1299808, 1893115, 2739248, 3938491, 5629407, 8000431, 11309295, 15904003, 22256183, 30998479, 42981170, 59337604
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^7+3*x^31+9*x^55+22*x^79+51*x^103+106*x^127+215*x^151+...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), this sequence (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(13/15) * exp(Pi*sqrt(26*n/15)) / (20*n). - Vaclav Kotesovec, Nov 28 2016

A160463 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 106.

Original entry on oeis.org

1, 4, 14, 40, 105, 249, 562, 1198, 2460, 4865, 9352, 17486, 31973, 57220, 100550, 173665, 295413, 495339, 819900, 1340655, 2167825, 3468579, 5495908, 8628080, 13428945, 20730689, 31757174, 48293585, 72933885, 109421095, 163135433, 241763735, 356246552
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^11+4*x^35+14*x^59+40*x^83+105*x^107+249*x^131+562*x^155+...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), this sequence (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^3/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(17/3) * exp(Pi*sqrt(34*n/15)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

A160506 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 106.

Original entry on oeis.org

1, 5, 20, 65, 190, 502, 1245, 2910, 6505, 13965, 29005, 58455, 114810, 220240, 413775, 762635, 1381550, 2463060, 4327445, 7500260, 12836645, 21712470, 36323930, 60143320, 98620425, 160238035, 258110955, 412367705, 653709340, 1028658150, 1607306688
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^15+5*x^39+20*x^63+65*x^87+190*x^111+502*x^135+1245*x^159+...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), this sequence (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^4/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(7/5) * exp(Pi*sqrt(14*n/5)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.

Original entry on oeis.org

1, 8, 44, 192, 726, 2457, 7648, 22220, 60993, 159478, 399906, 966600, 2261630, 5139897, 11378988, 24598683, 52033372, 107890610, 219630050, 439535138, 865784403, 1680352500, 3216454360, 6077280123, 11343018559, 20928404349, 38194869384, 68989715838
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			x^27+8*x^51+44*x^75+192*x^99+726*x^123+2457*x^147+7648*x^171+...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), this sequence (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^7/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

See Maple code in A160458 for formula.
a(n) ~ sqrt(11) * exp(Pi*sqrt(22*n/5)) / (2500*n). - Vaclav Kotesovec, Nov 28 2016

A278559 a(n) = A000041(25*n + 24).

Original entry on oeis.org

1575, 173525, 7089500, 169229875, 2841940500, 37027355200, 397125074750, 3646072432125, 29454549941750, 213636919820625, 1412749565173450, 8620496275465025, 49005643635237875, 261578907351144125, 1319510599727473500, 6324621482504294325, 28938037257084798150
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Examples

			a(4) = 5^2*63*A160460(4) + 5^5*52*A278555(3) + 5^7*63*A278556(2) + 5^10*6*A278557(1) + 5^12*A278558(0) = 771750 + 103512500 + 1028671875 + 1464843750 + 244140625 = 2841940500.

Links

Crossrefs

Cf. A000041, A160460, A213260, A278555, A278556, A278557, A278558.

Formula

a(n) = A213260(5*n + 4) = A000041(25*n + 24).
a(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
Showing 1-10 of 10 results.

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

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