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 1-3 of 3 results.

A285245 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 7, 10, 10, 10, 18, 24, 24, 24, 44, 56, 65, 65, 105, 129, 147, 147, 227, 292, 328, 355, 515, 645, 717, 771, 1107, 1367, 1562, 1670, 2429, 2949, 3339, 3555, 5073, 6181, 6961, 7546, 10582, 13059, 14619, 15789, 21925, 26886, 30235, 32575
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A001156, A006906, A285243, A285244.

Programs

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

Formula

a(n) ~ c * 2^(n/4), where
c = 6.362854320457366874306510139107365081972383711876544726... if mod(n,4)=0
c = 6.470997903106304472752360748461108347899808941622559468... if mod(n,4)=1
c = 6.154059402265470959096395812318265046714869376472639022... if mod(n,4)=2
c = 5.624747659153211728892605407048217108787120474872434485... if mod(n,4)=3

A285242 Expansion of Product_{k>=1} (1 + k*x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 4, 13, 9, 0, 0, 36, 36, 0, 16, 52, 63, 27, 64, 64, 108, 108, 64, 233, 277, 135, 27, 676, 676, 108, 204, 772, 1333, 765, 528, 420, 2628, 2628, 528, 1792, 3892, 3735, 1251, 5524, 5380, 4428, 4684, 6657, 12843, 10870, 6703, 3767, 28232
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A022629, A266891, A281790, A285240, A285243, A285244.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k, j]*k^j*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

A333653 Expansion of Product_{i>=1, j>=1} (1 + i*x^(i*j)).

Original entry on oeis.org

1, 1, 3, 7, 13, 27, 54, 98, 174, 335, 572, 1004, 1733, 2933, 4916, 8307, 13470, 22042, 35851, 57256, 91462, 145231, 227667, 355522, 554058, 853986, 1313121, 2010318, 3057827, 4627213, 6989808, 10481205, 15679549, 23365207, 34658909, 51241077, 75541695, 110852295, 162238415
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2020

Keywords

Links

Crossrefs

Cf. A022629, A107742, A285244, A318416, A332199.

Programs

  • Mathematica
    m = 38; CoefficientList[Series[Product[1 + i*x^(i*j), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(prod(i=1, N, prod(j=1, N\i, 1+i*x^(i*j))))
  • PARI
    N=40; x='x+O('x^N); Vec(prod(k=1, N, prod(d=1, k, 1+(k%d==0)*d*x^k)))

Formula

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} (1 + i*q^i).
G.f.: Product_{k>0} Product_{d|k} (1 + d*x^k).
Showing 1-3 of 3 results.

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

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