

A171238


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


6



1, 2, 5, 8, 16, 24, 40, 56, 88, 120, 176, 232, 328, 424, 576, 728, 968, 1208, 1568, 1928, 2464, 3000, 3768, 4536, 5632, 6728, 8248, 9768, 11864, 13960, 16784, 19608, 23400, 27192, 32192, 37192, 43760, 50328, 58824, 67320, 78280, 89240, 103200, 117160
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

A171238 also = 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 nth 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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000


FORMULA

Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is Lim_{n=1..inf.}, a leftshifted vector considered as a sequence.
From Wolfdieter Lang, Jul 15 2010: (Start)
O.g.f.: x*Q(x) with Q(x)*(1x)^2 = Q(x^2), for the eigensequence M*Q = Q with the column o.g.f.s (x^(2*m))/(1x)^2, m>=0, of M.
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(n1)b(n2), else b(n) = 2*b(n1) b(n2). (End)
G.f.: 1/((1x)*(1x^2)*(1x^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

CoefficientList[ Series[ 1/ Product[1  x^(2^i), {i, 0, 6}]^2, {x, 0, 60}], x] (* Robert G. Wilson v, May 11 2012 *)


CROSSREFS

Cf. A018819.
Cf. A000292
Sequence in context: A168470 A295998 A129299 * A096541 A226015 A137685
Adjacent sequences: A171235 A171236 A171237 * A171239 A171240 A171241


KEYWORD

nonn,easy


AUTHOR

Gary W. Adamson, Dec 05 2009


EXTENSIONS

More terms from Wolfdieter Lang, Jul 15 2010


STATUS

approved



