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 1-6 of 6 results.

A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 17, 26, 26, 26, 58, 94, 94, 94, 190, 298, 352, 352, 608, 896, 1112, 1112, 1752, 2641, 3289, 3559, 5095, 7499, 9227, 10307, 14051, 20111, 25520, 28760, 38843, 53467, 68191, 76831, 102187, 138283, 175543, 202813, 263905, 355220, 445364
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A285047, A266941, A285241, A285242, A285245.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1 - k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

Formula

a(n) ~ c * n * 2^(n/4), where
c = 37.4093119651465404809069752821426852731608123... if mod(n,4)=0
c = 37.6275180026872367633343656570058911570800766... if mod(n,4)=1
c = 37.7650387085085950514850376086515488784106690... if mod(n,4)=2
c = 37.4702467422193571732026074780460498930830447... if mod(n,4)=3

A285674 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^k.

Original entry on oeis.org

1, 1, 9, 36, 148, 489, 1959, 6326, 22741, 74072, 246436, 781189, 2523042, 7773342, 24200874, 73439472, 222247101, 660405663, 1958564056, 5715567301, 16623600991, 47780474694, 136623175876, 386983158080, 1090779014163, 3048348195528, 8478106666045
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 24 2017

Keywords

Links

Crossrefs

Cf. A077335, A266941, A285241, A285737.

Programs

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

Formula

a(n) ~ c * 3^(2*n/3) * n^2, where
c = 76631915822.1860553820452485980060616094557062528483009... if mod(n,3)=0
c = 76631915822.1819974623120987784506295282600132985390786... if mod(n,3)=1
c = 76631915822.1825610530012010285873110459423711856434442... if mod(n,3)=2
In closed form, a(n) ~ (Product_{k>=4}((1 - k^2/3^(2*k/3))^(-k)) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^2) + Product_{k>=4}((1 - (-1)^(2*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 4/3*(-1/3)^(1/3))^2 * (1 - (-1/3)^(2/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1)^(1/3)/3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^2))) * 3^(2*n/3) * n^2 / 54.

A285240 Expansion of Product_{k>=1} (1 + k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 8, 35, 115, 429, 1425, 4803, 15398, 48940, 151046, 459000, 1373219, 4037721, 11723911, 33566828, 94993571, 265722551, 735543433, 2015558930, 5471271099, 14719853084, 39266487114, 103908002173, 272855152096, 711272144097, 1841162650896, 4734074846631
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A022629, A027998, A266891, A285241, A285242.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 + k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]
nmax = 40; s = 1 + x; Do[s*=Sum[Binomial[k^2, j]*k^j*x^(j*k), {j, 0, Floor[nmax/k] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).

Original entry on oeis.org

1, 1, 17, 98, 514, 2435, 12752, 58849, 277362, 1243056, 5523734, 23889860, 102176581, 427458488, 1768064752, 7197695011, 28955246228, 114977761216, 451686925462, 1754581791860, 6749143188662, 25707194720502, 97041994691555, 363121143230292
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2017

Keywords

Links

Crossrefs

Column k=2 of A294585.
Cf. A077335, A285241, A285674, A294584.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 - k^2*x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 11 2017 *)
nmax = 30; s = 1 - x; Do[s *= Sum[Binomial[k^2, j]*(-1)^j*k^(2*j)*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^2*x^k)^k^2))

Formula

From Vaclav Kotesovec, Nov 14 2017: (Start)
a(n) ~ c * 3^(2*n/3) * n^8, where
if mod(n,3)=0 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576944425459411592809123023259309183199454386580509531344\
26216683391121761062030679551011342614958936988089343473390138...
if mod(n,3)=1 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943431618172412821798685989333734080090574886961583670\
65437558779530384541992249698997443314123905740649930258416583...
if mod(n,3)=2 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943586440772541471067224229278174424709431922476448338\
37991534958575385658058309282842532811502400165735702386411333...
In closed form, a(n) ~ ((Product_{k>=4} ((1 - k^2 / 3^(2*k/3))^(-k^2))) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^4) + (Product_{k>=4} ((1 - (-1)^(2*k/3) * k^2 / 3^(2*k/3))^(-k^2))) / ((-1)^(2*n/3) * (1 + 4/3 * (-1/3)^(1/3))^4 * (1 - (-1/3)^(2/3))) + (Product_{k>=4} ((1 - (-(-1)^(1/3))^k * k^2 / 3^(2*k/3))^(-k^2))) / ((-(-1)^(1/3))^n * (1 + (-1)^(1/3) / 3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^4)) * 3^(2*n/3) * n^8 / 793618560. - Vaclav Kotesovec, Nov 14 2017 (End)

A294588 Expansion of Product_{k>=1} (1 - k*x^k)^(k^2).

Original entry on oeis.org

1, -1, -8, -19, -13, 131, 497, 1149, 662, -4780, -22394, -55992, -75053, 44519, 614871, 2048356, 4311107, 4894537, -4917495, -44555890, -140584885, -292542844, -369256294, 110050979, 2247025840, 7757437455, 17600976064, 27240355017, 15747025855, -71354553868
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2017

Keywords

Links

Crossrefs

Column k=2 of A294587.
Cf. A285241.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k^2))

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 14, 1, 1, 17, 36, 42, 25, 1, 1, 33, 98, 140, 103, 56, 1, 1, 65, 276, 498, 481, 289, 97, 1, 1, 129, 794, 1844, 2419, 1774, 690, 198, 1, 1, 257, 2316, 7002, 12745, 12173, 5925, 1771, 354, 1, 1, 513, 6818, 27020, 69283, 89706, 56974, 20076, 4206, 672, 1
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   14, 42, 140, 498, 1844, ...

Links

Crossrefs

Columns k=0..3 give A006906, A266941, A285241, A294590.
Rows n=0-1 give A000012.
Cf. A144048, A294587.

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.
Showing 1-6 of 6 results.

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

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