The On-Line Encyclopedia of Integer Sequences (OEIS)

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A187154

Expansion of psi(x^4) / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
(history; published version)

#40 by Charles R Greathouse IV at Sun Feb 16 08:33:14 EST 2025

LINKS

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

Discussion

Sun Feb 16

08:33

OEIS Server: https://oeis.org/edit/global/3014

#39 by N. J. A. Sloane at Thu Jan 09 19:08:35 EST 2025

STATUS

proposed

approved

#38 by Peter Bala at Wed Jan 08 06:41:02 EST 2025

STATUS

editing

proposed

#37 by Peter Bala at Wed Jan 08 06:40:59 EST 2025

COMMENTS

Since phi(-x) = 1 + 2*Sum_{n k >= 1} (-1)^nk*x^(nk^2) == 1 (mod 2), it follows that the g.f. psi(x^4) / phi(-x) == psi(x^4) == Sum_{n k >= 0} x^(2*nk*(nk+1)) (mod 2). Hence a(n) is odd iff n = 2*k*(k + 1) for some nonegative integer k. - Peter Bala, Jan 07 2025

#36 by Peter Bala at Wed Jan 08 06:13:55 EST 2025

COMMENTS

Since phi(-x) = 1 + 2*Sum_{n >= 1} (-1)^n*x^(n^2) == 1 (mod 2), it follows that the g.f. psi(x^4) / phi(-x) == psi(x^4) == Sum_{n >= 0} x^(2*n(n+1)) (mod 2). Hence a(n) is odd iff n = 2*k*(k + 1) for some nonegative integer k. - Peter Bala, Jan 07 2025

#35 by Peter Bala at Tue Jan 07 16:56:23 EST 2025

COMMENTS

a(n) is odd iff n = 2*k*(k + 1) for some nonegative integer. - Peter Bala, Jan 07 2025

CROSSREFS

Cf. A001935, A001936, A0022568, A093085, A107035.

KEYWORD

nonn,easy

STATUS

approved

editing

#34 by Charles R Greathouse IV at Fri Mar 12 22:24:46 EST 2021

LINKS

M. Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

Discussion

Fri Mar 12

22:24

OEIS Server: https://oeis.org/edit/global/2897

#33 by N. J. A. Sloane at Wed Nov 13 21:54:14 EST 2019

LINKS

M. Somos, <a href="http://cis.csuohio.edu/~somosA010815/multiqa010815.pdftxt">Introduction to Ramanujan theta functions</a>

Discussion

Wed Nov 13

21:54

OEIS Server: https://oeis.org/edit/global/2830

#32 by Bruno Berselli at Tue Dec 05 04:07:32 EST 2017

STATUS

reviewed

approved

#31 by Michel Marcus at Tue Dec 05 00:44:45 EST 2017

STATUS

proposed

reviewed