A181649 An INVERT sequence for A010054.

1, 1, 2, 3, 6, 10, 18, 32, 57, 101, 179, 319, 566, 1006, 1786, 3174, 5638, 10016, 17793, 31609, 56153, 99753, 177211, 314810, 559255, 993501, 1764935, 3135366, 5569909, 9894819, 17577926, 31226796, 55473705, 98547807, 175067983, 311004383

Offset: 0

Author

Paul Barry, Nov 03 2010

Keywords

• easy
• nonn

Comments

Eigensequence of the sequence array for A010054.

Examples

			1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 18*x^6 + 32*x^7 + 57*x^8 + 101*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[1/(1 - q^(7/8)*eta[q^2]^2/eta[q]), {q, 0, 50}], q] (* G. C. Greubel, Sep 16 2018 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (1 - x * eta(x^2 + A)^2 / eta(x + A)), n))} /* Michael Somos, Jan 03 2013 */

Formula

G.f.: 1/(1-x*Product{k>0,(1 - x^(2k))/(1-x^(2k-1))}).

G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x^1 / (1 - x / (1 + x / (1 + x^2 / (1 - x / (1 + x / (1 + x^3 / (1 - x / (1 + x / ...)))))))))))). - Michael Somos, Jan 03 2013

a(n) ~ c / r^n, where r = 0.5629116358141452127351993944163442032777187438473224785071475357915... is the root of the equation (-1 + x)*QPochhammer(x^2, x^2) = QPochhammer(1/x, x^2) and c = 0.5730261147067572839709085685318242468812339379480160560847761872213851... - Vaclav Kotesovec, Jan 23 2024