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.

A321345 Expansion of 1/(1 - x) * Product_{k>=0} 1/(1 - x^(4^k))^(4^(k+1)).

Original entry on oeis.org

1, 5, 15, 35, 86, 206, 450, 890, 1751, 3411, 6401, 11405, 20076, 35036, 59876, 99156, 162345, 263821, 422871, 663691, 1031914, 1594610, 2440286, 3678886, 5504759, 8196659, 12117745, 17715581, 25744904, 37267624, 53652824, 76576760, 108763319, 153984019, 217058009
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2018

Keywords

Comments

Also the coefficient of x^(4*n) in the expansion of Product_{k>=0} 1/(1 - x^(4^k))^(4^k).

Examples

			Product_{k>=0} 1/(1 - x^(4^k))^(4^k) = 1 + x + x^2 + x^3 + 5*x^4 + 5*x^5 + 5*x^6 + 5*x^7 + 15*x^8 + 15*x^9 + 15*x^10 + 15*x^11 + 35*x^12 + 35*x^13 + 35*x^14 + 35*x^15 + ... .

Crossrefs

Cf. A321335, A321344.

Programs

  • PARI
    seq(n)={Vec(1/((1 - x)*prod(k=0, logint(n,4), (1 - x^(4^k) + O(x*x^n))^(4^(k+1)))))} \\ Andrew Howroyd, Nov 06 2018

A321354 Expansion of Product_{k>=0} (1 + x^(3^k))^(3^(k+1)).

Original entry on oeis.org

1, 3, 3, 10, 27, 27, 45, 108, 108, 147, 333, 333, 480, 1107, 1107, 1467, 3294, 3294, 3801, 8109, 8109, 9300, 19791, 19791, 22644, 48141, 48141, 50806, 104277, 104277, 107011, 216756, 216756, 224937, 458055, 458055, 454437, 905256, 905256, 870777, 1707075, 1707075
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2018

Keywords

Comments

Also the coefficient of x^(3*n) in the expansion of Product_{k>=0} (1 + x^(3^k))^(3^k).

Examples

			Product_{k>=0} (1 + x^(3^k))^(3^k) = 1 + x + 3*x^3 + 3*x^4 + 3*x^6 + 3*x^7 + 10*x^9 + 10*x^10 + 27*x^12 + 27*x^13 + 27*x^15 + 27*x^16 + 45*x^18 + 45*x^19 + ... .

Crossrefs

Cf. A073708, A321344, A321355, A321357.

A309046 Expansion of Product_{k>=0} (1 + x^(3^k) + x^(2*3^k) + x^(3^(k+1)))^(3^k).

Original entry on oeis.org

1, 1, 1, 4, 3, 3, 9, 6, 6, 25, 19, 19, 58, 39, 39, 105, 66, 66, 211, 145, 145, 394, 249, 249, 630, 381, 381, 1114, 733, 733, 1903, 1170, 1170, 2889, 1719, 1719, 4827, 3108, 3108, 7869, 4761, 4761, 11574, 6813, 6813, 18489, 11676, 11676, 28839, 17163, 17163, 41013, 23850
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 09 2019

Keywords

Comments

The trisection equals the three-fold convolution of this sequence with themselves.

Crossrefs

Cf. A054390, A237651, A309045, A321344.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[Product[(1 + x^(3^k) + x^(2 3^k) + x^(3^(k + 1)))^(3^k), {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 52; A[] = 1; Do[A[x] = (1 + x + x^2 + x^3) A[x^3]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=0} ((1 - x^(4*3^k))/(1 - x^(3^k)))^(3^k).
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3) * A(x^3)^3.
Showing 1-3 of 3 results.

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

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