A259094 - OEIS

%I #48 Nov 14 2024 23:22:53

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,2,2,2,1,1,1,1,2,2,3,

%T 3,1,1,1,1,2,2,3,4,3,1,1,1,1,2,2,3,4,4,3,1,1,1,1,2,2,3,4,5,5,4,1,1,1,

%U 1,2,2,3,4,5,6,6,4,1,1,1,1,2,2,3,4,5,6,7,7,4,1,1,1,1,2,2,3,4,5,6,8,9,8,5,1

%N From the Lecture Hall Theorem: array read by antidiagonals: T(n,k) = number of partitions of n into odd parts of size < 2k.

%C The Lecture Hall Theorem states that (the number of partitions (d1,d2,...,dn) of m such that 0 <= d1/1 <= d2/2 <= ... <= dn/n) equals (the number of partitions of m into odd parts less than 2n).

%H Seiichi Manyama, <a href="/A259094/b259094.txt">Antidiagonals n = 1..140, flattened</a>

%H M. Bousquet-Mélou, K. Eriksson, <a href="https://doi.org/10.1023/A:1009771306380">Lecture hall partitions</a>, The Ramanujan J. 1 (1997) 101-111.

%H M. Bousquet-Mélou, K. Eriksson, <a href="https://doi.org/10.1023/A:1009768118404">Lecture hall partitions II</a>, The Ramanujan J. 1 (1997) 165-185.

%H Mireille Bousquet-Mélou, Kimmo Eriksson, <a href="https://doi.org/10.1006/jcta.1998.2934">A Refinement of the Lecture Hall Theorem</a>, Journal of Combinatorial Theory, Series A, Volume 86, Issue 1, April 1999, Pages 63-84

%H Niklas Eriksen, <a href="http://www-igm.univ-mlv.fr/~fpsac/FPSAC02/ARTICLES/Eriksen2.pdf">A simple bijection between lecture hall partitions and partitions into odd integers</a> Formal Power Series and Algebraic Combinatorics. 2002.

%H Robin Whitty, <a href="http://www.theoremoftheday.org/NumberTheory/LectureHall/TotDLectureHall.pdf">The Lecture Hall Partition Theorem</a>

%H A. J. Yee, <a href="https://doi.org/10.1023/A:1012918510262">On combinatorics of lecture hall partitions</a>, The Ramanujan J. 5 (2001) 247-262.

%e The array begins:

%e   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ...

%e   1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14 ...

%e   1,1,1,2,2,3,4,4,5,6,7,8,9,10,11,13,14,15,17,18,20,22,23,25,27,29,31,33,35,37,40,42,44,47,49,52,55 ...

%e   1,1,1,2,2,3,4,5,6,7,9,10,12,14,16,19,21,24,27,30,34,38,42,46,51,56,61,67,73,79,86,93,100,108,116 ...

%e   1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58,65,74,82,91,102,113,124,138,151,165,182 ...

%e   1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227 ...

%e   1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,26,31,36,43,50,58,68,78,90,103,118,134,153,173,195,220,247,277 ...

%e   1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,37,45,52,61,72,83,96,111,128,146,168,191,217,247,279,314 ...

%e   ...

%e The successive antidiagonals are:

%e   [1]

%e   [1, 1]

%e   [1, 1, 1]

%e   [1, 1, 1, 1]

%e   [1, 1, 1, 2, 1]

%e   [1, 1, 1, 2, 2, 1]

%e   [1, 1, 1, 2, 2, 2, 1]

%e   [1, 1, 1, 2, 2, 3, 3, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 3, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 4, 3, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 5, 5, 4, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 5, 6, 6, 4, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 4, 1]

%e   [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 8, 5, 1]

%e   ...

%p G:=n->mul(1/(1-q^(2*i-1)),i=1..n);

%p M:=41;

%p G2:=n->seriestolist(series(G(n),q,M));

%p for n from 1 to 10 do lprint(G2(n)); od:

%p H:=n->[seq(G2(n-i+1)[i],i=1..n)];

%p for n from 1 to 14 do lprint(H(n)); od:

%t G[n_] := Product[1/(1-q^(2*i-1)), {i, 1, n}];

%t M = 41;

%t G2[n_] := CoefficientList[Series[G[n], {q, 0, M}], q];

%t For[n = 1, n <= 10, n++; Print[G2[n]]];

%t H[n_] := Table[G2[n-i+1][[i]], {i, 1, n}];

%t Reap[For[n = 1, n <= 14, n++, Print[H[n]]; Sow[H[n]]]][[2, 1]] // Flatten (* _Jean-François Alcover_, Jun 04 2017, translated from Maple *)

%Y Many rows of the array are already in the OEIS: A008620, A008672, A008673, A008674, A008675, A287997, A287998, A288000, A288001.

%K nonn,tabl

%O 1,14

%A _N. J. A. Sloane_, Jun 19 2015