A008672 -id:A008672 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 19 2015

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).

			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]
  ...

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.

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

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:

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 *)

A029032 Expansion of 1/((1-x)(1-x^3)(1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 27, 31, 35, 40, 45, 51, 57, 63, 70, 78, 86, 94, 103, 113, 123, 134, 145, 157, 170, 183, 197, 212, 227, 243, 260, 278, 296, 315, 335, 356, 378, 400, 423, 448, 473, 499, 526, 554, 583, 613, 644, 676, 709, 743

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 1, 3, 4, and 5. - David Neil McGrath, Sep 13 2014

Hoang Xuan Thanh, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,0,-1,-1,0,0,1,0,1,-1).

M := Matrix(13, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 3, 10, 12])) then 1 elif j=1 and member(i, [6, 7, 13]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..49); # Alois P. Heinz, Jul 25 2008

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)),{x,0,50}],x] (* Harvey P. Dale, Jan 04 2012 *)

a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=4, a(6)=5, a(7)=6, a(8)=8, a(9)=10, a(10)=12, a(11)=14, a(12)=17, a(n)=a(n-1)+a(n-3)-a(n-6)- a(n-7)+ a(n-10)+a(n-12)-a (n-13). - Harvey P. Dale, Jan 04 2012
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-1) = A008680(n).
a(n)-a(n-3) = A025772(n).
a(n)-a(n-4) = A008672(n).
a(n)-a(n-5) = A025767(n). (End)
a(n) = 1 + floor((2*n^3+39*n^2+228*n)/720). - Hoang Xuan Thanh, May 29 2025

A025799 Expansion of 1/((1-x^2)(1-x^3)(1-x^10)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 13, 11, 14, 13, 15, 14, 17, 15, 18, 17, 20, 18, 22, 20, 23, 22, 25, 23, 27, 25, 29, 27, 31, 29, 33, 31, 35, 33, 37, 35, 40, 37, 42, 40, 44, 42, 47, 44, 49, 47, 52, 49, 55, 52, 57, 55, 60, 57

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 2, 3, and 10. - Hoang Xuan Thanh, Aug 21 2025

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 2*x^11 + ...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,0,0,0,0,1,0,-1,-1,0,1).

Cf. A008672, A028242.

A025799[n_] := Floor[(n^2 + 15*n + 3*(-1)^n*(n + 7) + 99)/120];
Array[A025799, 100, 0] (* Paolo Xausa, Aug 25 2025 *)

{a(n) = if( n<-14, a(-15 - n), polcoeff( 1 / ((1 - x^2) * (1 - x^3) * (1 - x^10)) + x * O(x^n), n))}; /* Michael Somos, Mar 2003 */

{a(n) = n = (n - 3*(n%2)) / 2; (n^2 + 9*n)\30 + 1}; /* Michael Somos, Nov 16 2005 */

G.f.: 1/((1-x^2)(1-x^3)(1-x^10)).
a(n) = A008672( A028242(n - 2)). a(2*n + 3) = a(2*n) = A008672(n).- Michael Somos, Mar 2003
a(n) = a(-15 - n) for all n in Z. - Michael Somos, Nov 16 2005
a(n) = floor((n^2 + 15*n + 3*(n+7)*(-1)^n + 99)/120). - Hoang Xuan Thanh, Aug 21 2025

A211524 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x+5y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 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, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 95, 99, 103, 106, 110, 114, 118, 122, 126, 130, 134, 139

Offset: 0

Clark Kimberling, Apr 14 2012

nonn

For a guide to related sequences, see A211422.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 0, -1, 1).

Cf. A008672, A211422.

t[n_] := t[n] = Flatten[Table[-w + 3 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211524 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,0,0,0,0,0,0,1},69] (* Ray Chandler, Aug 02 2015 *)

a(n) = a(n-1)+a(n-3)-a(n-4)+a(n-5)-a(n-6)-a(n-8)+a(n-9).

G.f.: x^8/((1-x)*(1-x^3)*(1-x^5)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A029032 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)).

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Formula

A025799 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^10)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A211524 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x+5y.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A029032 Expansion of 1/((1-x)(1-x^3)(1-x^4)*(1-x^5)).

A025799 Expansion of 1/((1-x^2)(1-x^3)(1-x^10)).