A200751 Expansion of Product_{k>0} (1 - x^k)^(2^(k-1)) in powers of x.

1, -1, -2, -2, -3, -1, -2, 6, 12, 36, 74, 162, 301, 599, 1090, 1986, 3479, 5993, 9852, 15644, 23094, 30690, 31868, 9068, -82372, -345308, -1010956, -2577868, -6098822, -13751218, -29962588, -63604140, -132205949, -269982371, -542866266, -1076420666

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

Let F(a, x) = (1 - a) * (1 - a*x)^2 * (1 - a*x^2)^4 * ... where |x|<1/2. Then F(a, x) = (1 - a) * F(a*x, x)^2 and g.f. A(x) = F(x, x).

Euler transform of [ -1, -2, -4, -8, -16, ... ].

G.f.: (1 - x) * (1 - x^2)^2 * (1 - x^3)^4 * ...

Convolution inverse of A034691.

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A083413(k) * a(n-k). - Seiichi Manyama, Jul 17 2023

a(n) = Sum_{k=0..2n} (-1)^k*A360634(2n,k). - Alois P. Heinz, Sep 09 2023

Example

1 - x - 2*x^2 - 2*x^3 - 3*x^4 - x^5 - 2*x^6 + 6*x^7 + 12*x^8 + 36*x^9 + ...

Mathematica

a[n_] := SeriesCoefficient[Series[Product[(1 - x^k)^2^(k - 1),
{k, n}], {x, 0, n}], n]; Table[a[n], {n, 0, 35}] (* T. D. Noe, Nov 23 2011 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A) ^ 2^(k - 1)), n))}

Crossrefs

Cf. A034691, A083413, A360634.

Sequence in context: A282972 A354584 A088598 * A077083 A274528 A166248

Adjacent sequences: A200748 A200749 A200750 * A200752 A200753 A200754

Keyword

sign

Author

Michael Somos, Nov 21 2011

Status

approved