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.

A092119 EULER transform of A001511.

Original entry on oeis.org

1, 1, 3, 4, 10, 13, 26, 35, 66, 88, 150, 202, 331, 442, 688, 919, 1394, 1848, 2716, 3590, 5174, 6796, 9589, 12542, 17440, 22680, 31055, 40208, 54420, 70096, 93772, 120256, 159380, 203436, 267142, 339573, 442478, 560050, 724302, 913198, 1173375, 1473622
Offset: 0

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Author

Vladeta Jovovic, Mar 29 2004

Keywords

Comments

From Gary W. Adamson, Feb 11 2010: (Start)
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),
A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),
A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. (End)
Let M = triangle A173238 as an infinite lower triangular matrix. Then A092119 = lim_{n->infinity} M^n. Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + ...), and A(x) = polcoeff A092119. Then P(x) = A(x) / A(x^2), an example of a conjectured infinite set of operations (cf. A173238). - Gary W. Adamson, Feb 13 2010

Links

Crossrefs

Cf. A000041. - Gary W. Adamson, Feb 11 2010
Cf. A173241.

Programs

  • Maple
    # Uses EulerTransform from A358369.
t := EulerTransform(n -> padic[ordp](2*n, 2)):
seq(t(n), n = 0..41); # Peter Luschny, Nov 18 2022
  • Mathematica
    m = 42;
1/Product[QPochhammer[x^(2^k)], {k, 0, Log[2, m]//Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 14 2020, after Joerg Arndt *)
  • PARI
    N=66; x='x+O('x^N); /* that many terms */
gf=1/prod(e=0,ceil(log(N)/log(2)),eta(x^(2^e)));
Vec(gf) /* show terms */ /* Joerg Arndt, Jun 21 2011 */

Formula

G.f.: 1/Product_{k>=0} P(x^(2^k)) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011

A143374 G.f.: eta(q)*eta(q^3)*eta(q^9)*eta(q^27)*eta(q^81)*eta(q^243)*..., where eta(q) = Product((1-q^m), m=1..oo).

Original entry on oeis.org

1, -1, -1, -1, 1, 2, -1, 2, 0, -1, 0, 0, 0, -2, -2, 2, -3, -1, 1, 2, 3, 4, 1, -3, 0, -2, 3, -4, 2, 0, -1, -1, -2, -1, 0, -2, -2, 2, 2, -1, 0, 5, -1, 5, 0, 2, -3, -3, -3, 1, 3, 2, 2, -2, 4, -6, -4, 2, -2, -1, 2, -6, 0, 8, -4, -3, 2, 5, 1, -6, 3, 6, -1, 1, -4, -10, 1, 2, -1, 2, -5, -2, 6, 13, 4, 1, -1, 2, 1, 4, -4, -1
Offset: 0

Views

Author

N. J. A. Sloane and Gary W. Adamson, Feb 18 2010, Aug 14 2011

Keywords

Comments

eta(q) = A(q)/A(q^3), where A(q) is the g.f. for this sequence (cf. A010815).

Links

Crossrefs

Cf. A010815, A083650, A030189, A160832, A170925, A173241, A194087.

A173239 Triangle by columns, A000041 shifted down thrice, k>=0.

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 1, 7, 2, 11, 3, 1, 15, 5, 1, 22, 7, 2, 30, 11, 3, 1, 42, 15, 5, 1, 56, 22, 7, 2, 77, 30, 11, 3, 1, 101, 42, 15, 5, 1, 135, 56, 22, 7, 2, 176, 77, 30, 11, 3, 1, 231, 101, 42, 15, 5, 1, 297, 135, 56, 22, 7, 2, 385, 176, 77, 30, 11, 3, 1
Offset: 0

Views

Author

Gary W. Adamson, Feb 13 2010

Keywords

Comments

Row sums = A024787, the numbers of 3's in all partitions of n, where A024787 starts with offset 1: (0, 0, 1, 1, 2, 4, 6, 9, 15,...). Triangle A173239 row sums start with the first "1" of A024787.
Let the triangle = M as an infinite lower triangular matrix. Then Lim_{n->inf} = A173241, the Euler transform of A051064 (the ruler function for 3).
Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), then P(x) = A(x) / A(x^3), where A(x) = polcoeff A173241: (1 + x + 2x^2 + 4x^3 + 6x^4 + ...)
Refer to A173238 comments for three conjectures relating A000041 to the infinite set of generalized ruler function sequences.

Examples

			First few rows of the triangle =
1;
1;
2;
3, 1;
5, 1;
7, 2;
11, 3, 1;
15, 5, 1;
22, 7, 2;
30, 11, 3, 1;
42, 15, 5, 1;
56, 22, 7, 2;
77, 30, 11, 3, 1;
101, 42, 15, 5, 1;
135, 56, 22, 7, 2;
176, 77, 30, 11, 3, 1;
231, 101, 42, 15, 5, 1;
297, 135, 56, 22, 7, 2;
385, 176, 77, 30, 11, 3, 1;
490, 231, 101, 42, 15, 5, 1;
627, 297, 135, 56, 22, 7, 2;
792, 385, 176, 77, 30, 11, 3, 1;
1002,490, 231, 101, 42, 15, 5, 1;
1255, 627, 297, 135, 56, 22, 7, 2;
1575, 792, 385, 176, 77, 30, 11, 3, 1;
...

Crossrefs

Cf. A000041, A173238, A173241, A051064, A024787.

Formula

T(n,k) = A000041(n-3*k) for k=0..floor(n/3).

A373295 Expansion of 1/Product_{k>=1} (1 - x^k)^(valuation(k,4) + 1).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 373, 485, 649, 841, 1116, 1431, 1865, 2379, 3080, 3896, 4979, 6268, 7961, 9953, 12524, 15585, 19505, 24135, 29984, 36943, 45678, 56007, 68841, 84080, 102912, 125164, 152449, 184756, 224184, 270691, 327094, 393675
Offset: 0

Views

Author

Seiichi Manyama, May 31 2024

Keywords

Crossrefs

Cf. A092119, A173241, A373296, A373297, A373298.
Cf. A000041, A115362, A174065.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 4)+1)))

Formula

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 4^j)).
Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^4).

A373296 Euler transform of A055457.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 51, 69, 96, 129, 175, 235, 312, 410, 539, 700, 913, 1173, 1508, 1923, 2450, 3106, 3921, 4928, 6180, 7715, 9622, 11935, 14783, 18243, 22470, 27601, 33819, 41327, 50407, 61325, 74494, 90244, 109154, 131732, 158725, 190892, 229171, 274633, 328615
Offset: 0

Views

Author

Seiichi Manyama, May 31 2024

Keywords

Crossrefs

Cf. A092119, A173241, A373295, A373297, A373298.
Cf. A000041, A055457, A373219.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 5)+1)))

Formula

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 5^j)).
Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^5).

A373297 Euler transform of A373216.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 90, 118, 161, 212, 283, 367, 487, 624, 812, 1037, 1332, 1685, 2152, 2700, 3409, 4259, 5333, 6617, 8242, 10165, 12568, 15436, 18970, 23178, 28360, 34487, 41970, 50850, 61599, 74322, 89696, 107809, 129572, 155235, 185881, 221936
Offset: 0

Views

Author

Seiichi Manyama, May 31 2024

Keywords

Crossrefs

Cf. A092119, A173241, A373295, A373296, A373298.
Cf. A000041, A373216, A373220.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 6)+1)))

Formula

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 6^j)).
Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^6).

A373298 Euler transform of A373217.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 16, 23, 32, 45, 61, 84, 112, 152, 200, 265, 345, 451, 581, 750, 960, 1225, 1552, 1965, 2470, 3101, 3872, 4830, 5990, 7421, 9152, 11270, 13825, 16932, 20672, 25191, 30608, 37129, 44920, 54257, 65376, 78660, 94419, 113172, 135370, 161687, 192752
Offset: 0

Views

Author

Seiichi Manyama, May 31 2024

Keywords

Crossrefs

Cf. A092119, A173241, A373295, A373296, A373297.
Cf. A000041, A373217, A373221.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 7)+1)))

Formula

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 7^j)).
Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^7).
Showing 1-7 of 7 results.

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

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