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A171238 Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is Lim_{n=1..inf.} M^n. 6

%I

%S 1,2,5,8,16,24,40,56,88,120,176,232,328,424,576,728,968,1208,1568,

%T 1928,2464,3000,3768,4536,5632,6728,8248,9768,11864,13960,16784,19608,

%U 23400,27192,32192,37192,43760,50328,58824,67320,78280,89240,103200,117160

%N Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is Lim_{n=1..inf.} M^n.

%C A171238 also = polcoeff: (1,2,3,...)*(1,0,2,0,5,0,8,0,16,...).

%C Number of binary partitions of n into two kinds of parts. - _Joerg Arndt_, Feb 26 2015

%C Let the n-th convolution power of the sequence = B, with C = the aerated variant of B. It appears that B/C = the binomial sequence starting (1, 2n,...). Example: The sequence squared = (1, 4, 14, 36, 89, 192,...) = B; with C = (1, 0, 4, 0, 14, 0, 36,...). Then B/C = A000292: (1, 4, 10, 20, 35, 56,...). - _Gary W. Adamson_, Aug 15 2016

%H Vincenzo Librandi, <a href="/A171238/b171238.txt">Table of n, a(n) for n = 1..1000</a>

%F Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is Lim_{n=1..inf.}, a left-shifted vector considered as a sequence.

%F From _Wolfdieter Lang_, Jul 15 2010: (Start)

%F O.g.f.: x*Q(x) with Q(x)*(1-x)^2 = Q(x^2), for the eigensequence M*Q = Q with the column o.g.f.s (x^(2*m))/(1-x)^2, m>=0, of M.

%F Recurrence for b(n):=a(n+1): b(n)=0 if n<0, b(0)=1; if n even then b(n) = b(n/2)+2*b(n-1)-b(n-2), else b(n) = 2*b(n-1)- b(n-2). (End)

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^4)* ... *(1- x^(2^k))* ...)^2. - _Robert G. Wilson v_, May 11 2012

%F Convolution square of A018819. - _Michael Somos_, Mar 28 2014

%e G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 16*x^5 + 24*x^6 + 40*x^7 + 56*x^8 + ...

%t CoefficientList[ Series[ 1/ Product[1 - x^(2^i), {i, 0, 6}]^2, {x, 0, 60}], x] (* _Robert G. Wilson v_, May 11 2012 *)

%Y Cf. A018819.

%Y Cf. A000292

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Dec 05 2009

%E More terms from _Wolfdieter Lang_, Jul 15 2010

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Last modified March 21 18:59 EDT 2019. Contains 321382 sequences. (Running on oeis4.)