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.

Previous Showing 41-45 of 45 results.

A357475 Expansion of Product_{k>=1} 1 / (1 + x^k)^Fibonacci(k).

Original entry on oeis.org

1, -1, 0, -2, 0, -3, 0, -4, 2, -5, 8, 0, 26, 19, 74, 74, 195, 221, 464, 560, 1042, 1258, 2154, 2536, 3997, 4341, 6152, 5204, 5447, -1617, -10790, -39710, -83915, -181639, -336564, -633844, -1108334, -1952371, -3293590, -5568202, -9148916, -15017471, -24144556, -38697396, -61005748, -95708150
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Comments

Convolution inverse of A261050.

Crossrefs

Cf. A000045, A166861, A261050, A337009, A357179.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Product[1/(1 + x^k)^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d) d Fibonacci[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 45}]

Formula

a(n) = Sum_{k=0..n} (-1)^k * A337009(n,k). - Alois P. Heinz, Apr 30 2023

A358449 Euler transform of (0, 1, -2, 4, -8, 16, ...), (cf. A122803).

Original entry on oeis.org

1, 1, -1, 3, -4, 4, -2, 2, 2, -26, 80, -168, 351, -749, 1485, -2779, 5134, -9314, 16318, -27522, 44596, -68484, 96148, -113172, 77125, 122309, -750801, 2411307, -6424162, 15607886, -35846784, 79201548, -170009469, 356687423, -734287141, 1487086199, -2967980133
Offset: 0

Views

Author

Peter Luschny, Nov 17 2022

Keywords

Links

Crossrefs

Cf. A034691, A122803.

Programs

  • Maple
    # Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(-2, 0)): seq(a(n), n=0..36);
  • Sage
    # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-2, 0)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A357179 Expansion of Product_{k>=1} (1 - x^k)^Fibonacci(k).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 2, 1, 5, 6, 14, 15, 32, 40, 64, 86, 131, 166, 237, 287, 362, 389, 368, 149, -339, -1477, -3680, -7827, -15245, -28270, -50493, -87886, -149827, -250966, -414542, -675741, -1089267, -1736640, -2741788, -4284837, -6632751, -10162683, -15412613, -23110653, -34236290
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Comments

Convolution inverse of A166861.

Crossrefs

Cf. A000045, A166861, A261050, A357475.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Product[(1 - x^k)^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d Fibonacci[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 45}]

A291650 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))^Fibonacci(k).

Original entry on oeis.org

1, 1, 2, 5, 4, 12, 14, 16, 42, 35, 65, 100, 84, 205, 201, 254, 490, 386, 749, 917, 851, 1816, 1566, 2260, 3513, 2784, 5566, 5748, 6116, 11366, 9048, 14740, 19037, 16095, 31576, 28505, 35218, 56334, 43671, 77512, 85163, 80577, 147756, 121016, 172408, 236022
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Number of partitions of n into distinct Fibonacci parts (1 counted as single Fibonacci number), where Fibonacci(k) different parts of size Fibonacci(k) are available (1a, 2a, 2b, 3a, 3b, 3c, ...).

Examples

			a(3) = 5 because we have [3a], [3b], [3c], [2a, 1a] and [2b, 1a].

Links

Crossrefs

Cf. A000045, A000119, A000121, A026007, A166861, A200544, A239002, A260787, A261050.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^Fibonacci[k])^Fibonacci[k], {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^A000045(k))^A000045(k).

A308446 Expansion of Product_{k>=1} 1/(1 - x^k)^Fibonacci(2*k).

Original entry on oeis.org

1, 1, 4, 12, 39, 118, 371, 1129, 3468, 10524, 31910, 96155, 289016, 865000, 2581577, 7679762, 22784896, 67418329, 199004329, 586052299, 1722165404, 5050349249, 14781877481, 43185726143, 125949155473, 366716549379, 1066057177765, 3094398005409, 8969054893842
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2019

Keywords

Comments

Euler transform of A001906.

Crossrefs

Cf. A000045, A001906, A032170, A034691, A088305, A166861.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^Fibonacci[2 k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Fibonacci[2 d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

Formula

a(n) ~ phi^(2*n) * exp(2*sqrt(n)/5^(1/4) - 3/10 + S) / (2 * 5^(1/8) * sqrt(Pi) * n^(3/4)), where S = Sum_{k>=2} 1/((phi^(2*k) - 3 + 1/phi^(2*k))*k) = 0.155349347463140787939176213528043741704916536093946010733676987281... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 28 2019
Previous Showing 41-45 of 45 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.