A241204 Expansion of (1 + 2*x)^2/(1 - 2*x)^2.

1, 8, 32, 96, 256, 640, 1536, 3584, 8192, 18432, 40960, 90112, 196608, 425984, 917504, 1966080, 4194304, 8912896, 18874368, 39845888, 83886080, 176160768, 369098752, 771751936, 1610612736, 3355443200, 6979321856, 14495514624, 30064771072, 62277025792

Offset: 0

Author

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 17 2014

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Subsequence of A008574.

Programs

Magma

R:=PowerSeriesRing(Integers(), 41); Coefficients(R!((1+2*x)^2/(1-2*x)^2));

Maple

A241204:= n->`if`(n=0, 1, 2^(n+2)*n); seq(A241204(n), n=0..20); # Wesley Ivan Hurt, Apr 22 2014

Mathematica

Table[2^(n+2)*n + Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 07 2023 *)
LinearRecurrence[{4,-4},{1,8,32},30] (* Harvey P. Dale, Jun 23 2025 *)

PARI

Vec((2*x+1)^2/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Apr 22 2014

Sage

def A241204(i):
    if i==0: return 1
    else: return 2^(2+i)*i;
[A241204(n) for n in (0..30)] # Bruno Berselli, Apr 23 2014

Formula

a(n) = 2^(2+n)*n for n>0. - Colin Barker, Apr 23 2014

a(n) = 4*a(n-1)-4*a(n-2) for n>2. - Colin Barker, Apr 23 2014

From Amiram Eldar, Jan 13 2021: (Start)

Sum_{n>=1} 1/a(n) = log(2)/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2)/4. (End)

E.g.f.: 1 + 8*x*exp(x). - G. C. Greubel, Jun 07 2023