The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000009

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Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.
(history; published version)

#529 by Charles R Greathouse IV at Sun Feb 16 08:32:18 EST 2025

LINKS

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>, <a href="http://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function bk</a>, <a href="http://mathworld.wolfram.com/EulerIdentity.html">Euler Identity</a>, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.

Discussion

Sun Feb 16

08:32

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

#528 by N. J. A. Sloane at Thu Jan 09 19:06:53 EST 2025

STATUS

proposed

#527 by Peter Bala at Wed Jan 08 06:36:05 EST 2025

STATUS

editing

#526 by Peter Bala at Wed Jan 08 06:35:57 EST 2025

COMMENTS

The g.f. Product_{k >= 0} 1 + x^k = Product_{k >= 0} 1 - x^k + 2*x^k == Product_{k >= 0} 1 - x^k == Sum_{n in Z} (-1)^n*x^(n*(3*n-1)/2) (mod 2) by Euler's pentagonal number theorem. It follows that a(n) is odd iff n = k*(3*k - 1)/2 for some integer k, i.e., iff n is a generalized pentagonal number A001318. - Peter Bala, Jan 07 2025

#525 by Peter Bala at Wed Jan 08 05:26:32 EST 2025

COMMENTS

The g.f. Product_{k >= 0} 1 + x^k = Product_{k >= 0} 1 - x^k + 2*x^k == Product_{k >= 0} 1 - x^k == Sum_{n in Z} (-1)^n*x^(n*(3*n-1)/2) (mod 2) by Euler's pentagonal number theorem. It follows that a(n) is odd iff n = k*(3*k+1)/2 for some integer k, i.e., iff n is a generalized pentagonal number A001318. - Peter Bala, Jan 07 2025

#524 by Peter Bala at Wed Jan 08 05:23:08 EST 2025

COMMENTS

a(n) is odd iff n is a generalized pentagonal number A001318, i.e., iff n = k*(3*k+1)/2 for some integer k. - Peter Bala, Jan 07 2025

#523 by Peter Bala at Tue Jan 07 17:38:06 EST 2025

COMMENTS

STATUS

approved

#522 by Peter Luschny at Thu Dec 05 11:11:31 EST 2024

STATUS

proposed

#521 by Michael De Vlieger at Thu Dec 05 11:09:23 EST 2024

STATUS

editing

#520 by Michael De Vlieger at Thu Dec 05 11:09:18 EST 2024

LINKS

STATUS

approved