A092119 EULER transform of A001511.

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

0,3

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

N. J. A. Sloane, Transforms

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

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 *)

Prog

(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 */

Crossrefs

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

Cf. A173241.

Sequence in context: A073443 A257494 A302347 * A362649 A143372 A035594

Adjacent sequences: A092116 A092117 A092118 * A092120 A092121 A092122

Keyword

nonn

Author

Vladeta Jovovic, Mar 29 2004

Status

approved