OFFSET
1,2
COMMENTS
Also equals polcoeff: (1,2,3,...)*(1,0,2,0,5,0,8,0,16,...).
Number of binary partitions of n into two kinds of parts. - Joerg Arndt, Feb 26 2015
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
LINKS
Georg Fischer, Table of n, a(n) for n = 1..1000 [first 128 terms from Vincenzo Librandi]
FORMULA
Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is lim_{k->infinity}, a left-shifted vector considered as a sequence.
From Wolfdieter Lang, Jul 15 2010: (Start)
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.
Recurrence for b(n):=a(n+1): b(n)=0 if n < 0, b(0)=1; if n is even then b(n) = b(n/2) + 2*b(n-1) - b(n-2), otherwise b(n) = 2*b(n-1) - b(n-2). (End)
G.f.: 1/((1-x)*(1-x^2)*(1-x^4)* ... *(1- x^(2^k))* ...)^2. - Robert G. Wilson v, May 11 2012
Convolution square of A018819. - Michael Somos, Mar 28 2014
EXAMPLE
G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 16*x^5 + 24*x^6 + 40*x^7 + 56*x^8 + ...
MATHEMATICA
imax=10; CoefficientList[ Series[ 1/ Product[1 - x^(2^i), {i, 0, imax}]^2, {x, 0, 2^imax}], x] (* Robert G. Wilson v, May 11 2012; range of "i" amended by Georg Fischer, May 12 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Dec 05 2009
EXTENSIONS
More terms from Wolfdieter Lang, Jul 15 2010
STATUS
approved