The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A307059

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A307059

Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).
(history; published version)

#20 by N. J. A. Sloane at Wed Jan 24 10:09:01 EST 2024

STATUS

proposed

approved

#19 by G. C. Greubel at Wed Jan 24 01:40:13 EST 2024

STATUS

editing

proposed

#18 by G. C. Greubel at Wed Jan 24 01:40:09 EST 2024

FORMULA

G.f.: 1/(2 - QPochhammer(n,x,x)). - G. C. Greubel, Sep 08 2023

CROSSREFS

Cf. A307057, A307058, A307060, A307062, A307063.

STATUS

approved

editing

#17 by OEIS Server at Fri Sep 08 04:05:19 EDT 2023

LINKS

G. C. Greubel, <a href="/A307059/b307059_1.txt">Table of n, a(n) for n = 0..2500</a>

#16 by Joerg Arndt at Fri Sep 08 04:05:19 EDT 2023

STATUS

reviewed

approved

Discussion

Fri Sep 08

04:05

OEIS Server: Installed first b-file as b307059.txt.

#15 by Michel Marcus at Fri Sep 08 03:51:40 EDT 2023

STATUS

proposed

reviewed

#14 by G. C. Greubel at Fri Sep 08 02:47:08 EDT 2023

STATUS

editing

proposed

#13 by G. C. Greubel at Fri Sep 08 02:45:44 EDT 2023

COMMENTS

Alternating row sums of Riordan triangle (1, 1 -( Product_{j>=1} - (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - Wolfdieter Lang, Feb 17 2021

LINKS

G. C. Greubel, <a href="/A307059/b307059_1.txt">Table of n, a(n) for n = 0..2500</a>

#12 by G. C. Greubel at Fri Sep 08 02:29:46 EDT 2023

FORMULA

G.f.: 1/(2 - QPochhammer(n,x,x)). - G. C. Greubel, Sep 08 2023

MATHEMATICA

nmax = 65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

PROG

(Magma)

m:=80;

R<x>:=PowerSeriesRing(Integers(), m);

Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // G. C. Greubel, Sep 08 2023

(SageMath)

from sage.modular.etaproducts import qexp_eta

m=80;

def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )

def A307059_list(prec):

P.<x> = PowerSeriesRing(QQ, prec)

return P( f(x) ).list()

A307059_list(m) # G. C. Greubel, Sep 08 2023

STATUS

approved

editing

#11 by Peter Luschny at Wed Feb 17 06:40:12 EST 2021

STATUS

reviewed

approved