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.

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

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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)

A301335 a(n) = [x^n] 1/(1 + (1/2)*n*(1 - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 27, 260, 3175, 47304, 833147, 16941120, 390611331, 10070060200, 287028156162, 8962583345856, 304255011200647, 11156593415089808, 439452231820920000, 18505340390664634384, 829599437871129843839, 39447684087807950938908, 1983038000428208822539998, 105080571577382659860160800
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 18 2018

Keywords

Comments

Number of compositions (ordered partitions) of n into squares of n kinds.

Links

Crossrefs

Cf. A000290, A006456, A240944, A300974, A301334.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + (1/2) n (1 - EllipticTheta[3, 0, x])), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - n Sum[x^k^2, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^2)).
a(n) ~ n^n * (1 + 1/n^2 - 3/n^3 + 1/(2*n^4) - 13/(2*n^5) + 127/(6*n^6) - 4/n^7 + 335/(8*n^8) - 665/(4*n^9) + 337/(15*n^10) + ...). - Vaclav Kotesovec, Mar 19 2018

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 88, 192, 420, 922, 2024, 4440, 9736, 21352, 46832, 102720, 225298, 494144, 1083804, 2377112, 5213736, 11435312, 25081112, 55010496, 120654744, 264632554, 580419672, 1273036832, 2792156864, 6124049048, 13431901808, 29460245120, 64615275940
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Crossrefs

Cf. A000122, A004402, A006456, A025192, A032803, A240944, A279225, A279226, A280542, A320654.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - 2 Sum[x^(k^2), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Formula

G.f.: 1 / (2 - theta_3(x)), where theta_3() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A000122(k) * a(n-k).
Showing 1-3 of 3 results.

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