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.

A294622 Number of partitions of n into generalized octagonal numbers (A001082).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 8, 8, 9, 9, 10, 13, 13, 14, 16, 17, 20, 20, 21, 24, 25, 28, 31, 33, 36, 37, 40, 45, 47, 50, 55, 59, 65, 67, 70, 77, 81, 87, 94, 99, 107, 111, 117, 127, 133, 141, 152, 160, 172, 178, 186, 201, 210, 223, 237, 249, 267, 276, 289, 308, 322, 341, 360, 378, 401
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2017

Keywords

Examples

			a(8) = 3 because we have [8], [5, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A001082, A007294, A095699, A279041, A294621, A294624.

Programs

  • Mathematica
    nmax = 74; CoefficientList[Series[Product[1/((1 - x^(k (3 k - 2))) (1 - x^(k (3 k + 2)))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/((1 - x^(k*(3*k-2)))*(1 - x^(k*(3*k+2)))).

A294654 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(5*k-3)/2)*(1 - x^(2*k))^(k*(5*k+3)/2)).

Original entry on oeis.org

1, 1, 5, 12, 35, 81, 208, 475, 1123, 2505, 5617, 12192, 26368, 55797, 117255, 242660, 498126, 1010515, 2033662, 4053214, 8017622, 15729219, 30643069, 59268267, 113898873, 217480476, 412813600, 779042099, 1462188257, 2729852845, 5070966794, 9373909586, 17247473718
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2017

Keywords

Comments

Euler transform of the generalized heptagonal numbers (A085787).

Links

Crossrefs

Cf. A000294, A085787, A278769, A294591, A294621, A294655.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (5 k - 3)/2) (1 - x^(2 k))^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A085787(k).
a(n) ~ exp(Pi * (2/3)^(5/4) * n^(3/4) + 5*Zeta(3) * sqrt(3*n) / (2^(3/2) * Pi^2) - (75*3^(1/4) * Zeta(3)^2 / (2^(13/4) * Pi^5) + Pi / (2^(17/4) * 3^(3/4))) * n^(1/4) + 375 * Zeta(3)^3 / (8*Pi^8) - 5*Zeta(3) / (64*Pi^2) + 1/12) * Pi^(1/12) / (A * 2^(11/6) * 3^(7/48) * n^(31/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017

A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 2, 3, 1, 0, 3, 3, 1, 2, 2, 1, 1, 3, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 5, 3, 1, 3, 4, 3, 4, 5, 2, 3, 5, 4, 3, 4, 5, 4, 4, 3, 5, 5, 3, 5, 7, 5, 3, 6, 6, 6, 6, 5, 5, 6, 6, 5, 8, 7, 5, 5, 6, 7, 8, 8
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2017

Keywords

Examples

			a(18) = 2 because we have [18] and [13, 4, 1].

Links

Crossrefs

Cf. A024940, A085787, A279280, A290942, A294621, A294624.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(k (5 k - 3)/2)) (1 + x^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2))*(1 + x^(k*(5*k+3)/2)).
Showing 1-3 of 3 results.

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