A171238 Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is lim_{k->infinity} M^k.

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

Offset: 1

Author

Gary W. Adamson, Dec 05 2009

Keywords

• nonn

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

Examples

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

Links

• Georg Fischer, Table of n, a(n) for n = 1..1000 [first 128 terms from Vincenzo Librandi]

Crossrefs

Cf. A000292, A018819, A122196.

Programs

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 *)

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

Extensions

More terms from Wolfdieter Lang, Jul 15 2010