The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A237757 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A237757

Number of partitions of n such that 2*(least part) = (number of parts).
(history; published version)

#17 by Vaclav Kotesovec at Sat Jan 22 07:31:08 EST 2022

STATUS

editing

approved

#16 by Vaclav Kotesovec at Sat Jan 22 07:31:01 EST 2022

FORMULA

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(7/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jan 22 2022

STATUS

approved

editing

#15 by N. J. A. Sloane at Tue Feb 02 12:07:59 EST 2021

STATUS

editing

approved

#14 by N. J. A. Sloane at Tue Feb 02 12:07:56 EST 2021

MAPLE

#Alternative program based on conjectural g.f.

G := add(x^(2*(n+1)^2)/mul(1 - x^k, k=1..2*n+1), n = 0..6):

S := series(G, x, 101):

seq(coeff(S, x, j), j = 1..100); # Peter Bala, Feb 02 2021

STATUS

proposed

editing

#13 by Peter Bala at Tue Feb 02 11:33:12 EST 2021

STATUS

editing

proposed

Discussion

Tue Feb 02

11:35

Bruno Berselli: Dear Peter, we shouldn't use conjecture in programs.

12:07

N. J. A. Sloane: Bruno is right. No programs based on conjectures are allowed.  They get misused.

#12 by Peter Bala at Tue Feb 02 06:39:13 EST 2021

FORMULA

Conjectural g.f.: Sum_{n >= 0} q^(2*(n+1)^2)/Product_{k = 1..2*n+1} 1 - q^k. - Peter Bala, Feb 02 2021

MAPLE

#Alternative program based on conjectural g.f.

G := add(x^(2*(n+1)^2)/mul(1 - x^k, k=1..2*n+1), n = 0..6):

S := series(G, x, 101):

seq(coeff(S, x, j), j = 1..100); # Peter Bala, Feb 02 2021

STATUS

approved

editing

#11 by N. J. A. Sloane at Wed Jul 01 20:23:51 EDT 2020

STATUS

proposed

approved

#10 by Robert Israel at Wed Jul 01 13:47:41 EDT 2020

STATUS

editing

proposed

#9 by Robert Israel at Wed Jul 01 13:47:27 EDT 2020

LINKS

Robert Israel, <a href="/A237757/b237757.txt">Table of n, a(n) for n = 1..10000</a>

MAPLE

f:= proc(n) local t, k, np;

t:= 0;

for k from 1 do

np:= n - 1 - 2*k*(k-1);

if np < 2*k-1 then return t fi;

t:= t + combinat:-numbpart(np, 2*k-1) - combinat:-numbpart(np, 2*k-2)

od;

end proc:

map(f, [$1..100]); # Robert Israel, Jul 01 2020

STATUS

approved

editing

#8 by Alois P. Heinz at Thu Aug 17 16:05:27 EDT 2017

STATUS

proposed

approved