A092119 - OEIS

%I #30 Nov 18 2022 09:02:34

%S 1,1,3,4,10,13,26,35,66,88,150,202,331,442,688,919,1394,1848,2716,

%T 3590,5174,6796,9589,12542,17440,22680,31055,40208,54420,70096,93772,

%U 120256,159380,203436,267142,339573,442478,560050,724302,913198,1173375,1473622

%N EULER transform of A001511.

%C From _Gary W. Adamson_, Feb 11 2010: (Start)

%C Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),

%C A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),

%C 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)

%C 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

%H Seiichi Manyama, <a href="/A092119/b092119.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

%p # Uses EulerTransform from A358369.

%p t := EulerTransform(n -> padic[ordp](2*n, 2)):

%p seq(t(n), n = 0..41); # _Peter Luschny_, Nov 18 2022

%t m = 42;

%t 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_ *)

%o (PARI)  N=66; x='x+O('x^N); /* that many terms */

%o gf=1/prod(e=0,ceil(log(N)/log(2)),eta(x^(2^e)));

%o Vec(gf) /* show terms */ /* _Joerg Arndt_, Jun 21 2011 */

%Y Cf. A000041. - _Gary W. Adamson_, Feb 11 2010

%Y Cf. A173241.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Mar 29 2004