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 1-7 of 7 results.

A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 4, 3, 0, 0, 6, 6, 0, 4, 7, 6, 3, 8, 8, 6, 6, 4, 21, 20, 4, 1, 34, 34, 2, 8, 23, 44, 28, 19, 18, 54, 54, 18, 56, 65, 46, 25, 100, 94, 38, 42, 85, 169, 107, 56, 69, 226, 194, 62, 111, 194, 241, 125, 215, 246, 258, 207, 283, 437, 292
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 14 2017

Keywords

Links

Crossrefs

Cf. A000219, A026007, A027998, A033461, A255528, A266891, A284896, A285047.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s*=Sum[Binomial[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]

Formula

a(n) ~ exp(sqrt(n/6)*Pi) / (2^(11/6) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 15 2017

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 15, 24, 24, 24, 44, 80, 80, 80, 131, 221, 266, 266, 386, 566, 746, 746, 990, 1474, 1924, 2089, 2529, 3709, 4609, 5269, 6130, 8576, 11096, 12746, 14937, 19397, 25697, 28997, 34111, 43135, 56365, 65905, 76219, 95770, 120070, 144370, 163661
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2017

Keywords

Links

Crossrefs

Cf. A001156, A033461, A281790, A285047, A291649, A291696, A291720.

Programs

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

Formula

log(a(n)) ~ 5 * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5) / (2^(6/5) * 3^(3/5)).

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

Original entry on oeis.org

1, 1, 3, 10, 39, 151, 588, 2304, 9111, 36307, 145553, 586246, 2370264, 9614242, 39105580, 159444160, 651468967, 2666771488, 10934393619, 44899828056, 184616878289, 760010818689, 3132147583744, 12921037206764, 53351800567200, 220478125956426, 911839751015196, 3773836780169050
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2018

Keywords

Comments

Number of partitions of n into squares of n kinds.

Links

Crossrefs

Cf. A000290, A001156, A008485, A023871, A240944, A279225, A285047, A300975, A301518.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^2*j, i-1), j=0..n/i^2)))
      end: b(n, isqrt(n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018
  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k^2)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

Formula

From Vaclav Kotesovec, Mar 23 2018: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 4.216358447600641565890184638418336163396695730036... and
c = 0.26442245016754864773722176155288663999776... (End)

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

A291667 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^k.

Original entry on oeis.org

1, 2, 2, 2, 6, 10, 10, 10, 18, 32, 38, 38, 50, 86, 110, 110, 134, 206, 272, 290, 342, 466, 610, 682, 770, 1012, 1310, 1492, 1654, 2130, 2698, 3066, 3410, 4210, 5310, 6106, 6812, 8078, 10118, 11750, 13006, 15198, 18654, 21810, 24178, 28092, 33682, 39330, 43866
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 29 2017

Keywords

Comments

Convolution of A285047 and A281790.

Links

Crossrefs

Cf. A001156, A033461, A285047, A281790, A291666.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[((1+x^(k^2))/(1-x^(k^2)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

log(a(n)) ~ Pi*sqrt(n/2).

A303168 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, 13, 13, 13, 13, 13, 18, 18, 18, 24, 24, 24, 24, 24, 30, 30, 30, 39, 39, 39, 39, 39, 46, 46, 46, 58, 58, 58, 64, 64, 72, 72, 72, 87, 87, 87, 99, 99, 112, 112, 112, 130, 130, 130, 148, 148, 166, 166, 166, 187
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Comments

Number of partitions of n into 1 kind of part 1, 2 kinds of part 8, 3 kinds of part 27, ..., k kinds of part k^3.

Links

Crossrefs

Cf. A000219, A000578, A003108, A023872, A285047, A291720, A298434, A300975.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 - x^k^3)^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A000578(k))^k.

A320846 Expansion of Product_{k>=1} 1/(1 - x^(k^2))^A037444(k).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 6, 10, 10, 10, 14, 22, 22, 22, 35, 47, 57, 57, 79, 95, 115, 115, 146, 217, 247, 267, 307, 433, 473, 513, 598, 779, 985, 1045, 1253, 1489, 1861, 1941, 2272, 2859, 3397, 3847, 4301, 5467, 6171, 6991, 7688, 9531, 11559, 12749, 14693
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 11 2018

Keywords

Comments

a(n) is the number of partitions of n into squares k^2 of A037444(k) kinds.

Examples

			a(8) = 6 because we have [{4}, {4}], [{4}, {1, 1, 1, 1}], [{4}, {1}, {1}, {1}, {1}], [{1, 1, 1, 1}, {1, 1, 1, 1}], [{1, 1, 1, 1}, {1}, {1}, {1}, {1}] and [{1}, {1}, {1}, {1}, {1}, {1}, {1}, {1}].

Links

Crossrefs

Cf. A000290, A001156, A001970, A037444, A045842, A285047, A300300.

Programs

  • Mathematica
    b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^k^2), {k, 1, n}], {x, 0, n^2}]; a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^k^2)^b[k], {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 0, 52}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A000290(k))^A001156(A000290(k)).
Showing 1-7 of 7 results.

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

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