A259094 From the Lecture Hall Theorem: array read by antidiagonals: T(n,k) = number of partitions of n into odd parts of size < 2k.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Jun 19 2015

Keywords

• nonn
• tabl

Comments

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).

Examples

			The array begins:
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  ...
The successive antidiagonals are:
  [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, 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, 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]
  ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..140, flattened
• M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions, The Ramanujan J. 1 (1997) 101-111.
• M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, The Ramanujan J. 1 (1997) 165-185.
• Mireille Bousquet-Mélou, Kimmo Eriksson, A Refinement of the Lecture Hall Theorem, Journal of Combinatorial Theory, Series A, Volume 86, Issue 1, April 1999, Pages 63-84
• Niklas Eriksen, A simple bijection between lecture hall partitions and partitions into odd integers Formal Power Series and Algebraic Combinatorics. 2002.
• Robin Whitty, The Lecture Hall Partition Theorem
• A. J. Yee, On combinatorics of lecture hall partitions, The Ramanujan J. 5 (2001) 247-262.

Crossrefs

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

Programs

Maple

G:=n->mul(1/(1-q^(2*i-1)),i=1..n);
M:=41;
G2:=n->seriestolist(series(G(n),q,M));
for n from 1 to 10 do lprint(G2(n)); od:
H:=n->[seq(G2(n-i+1)[i],i=1..n)];
for n from 1 to 14 do lprint(H(n)); od:

Mathematica

G[n_] := Product[1/(1-q^(2*i-1)), {i, 1, n}];
M = 41;
G2[n_] := CoefficientList[Series[G[n], {q, 0, M}], q];
For[n = 1, n <= 10, n++; Print[G2[n]]];
H[n_] := Table[G2[n-i+1][[i]], {i, 1, n}];
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 *)