cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

A335724 a(n) is the number of smallest parts in the overpartitions of n.

Original entry on oeis.org

2, 6, 12, 26, 44, 84, 136, 230, 366, 580, 884, 1356, 2012, 2968, 4320, 6226, 8856, 12522, 17508, 24324, 33528, 45892, 62392, 84372, 113374, 151548, 201552, 266752, 351380, 460920, 601992, 783158, 1014984, 1310600, 1686408, 2162814, 2764748, 3523324, 4476720, 5671748
Offset: 1

Views

Author

Jeremy Lovejoy, Jun 19 2020

Keywords

Examples

			There are 14 overpartitions of 4: [4], [4'], [3,1], [3,1'], [3',1], [3',1'], [2,2], [2',2], [2,1,1], [2,1',1], [2',1,1], [2',1',1], [1,1,1,1], [1',1,1,1], and so a(4) = 26.

Links

Crossrefs

Cf. A015128, A092269, A235792, A335728, A335730.

Formula

G.f.: 2*(Product_{k>=1} (1+q^k)/(1-q^k))*Sum_{n>=1} (q^n*Product_{j=1..n}(1-q^j))/((1-q^n)^2*Product_{j=1..n}(1+q^j)).
a(n) = A335728(n) + A335730(n).

A335728 a(n) is the number of smallest parts in the overpartitions of n having even smallest part.

Original entry on oeis.org

0, 2, 0, 6, 4, 12, 12, 30, 36, 60, 80, 132, 180, 264, 360, 522, 712, 990, 1344, 1844, 2472, 3324, 4420, 5892, 7764, 10212, 13344, 17400, 22556, 29160, 37524, 48166, 61560, 78456, 99648, 126234, 159396, 200740, 252096, 315828
Offset: 1

Views

Author

Jeremy Lovejoy, Jun 19 2020

Keywords

Examples

			There are 14 overpartitions of 4: [4], [4'], [3,1], [3,1'], [3',1], [3',1'], [2,2], [2',2], [2,1,1], [2,1',1], [2',1,1], [2',1',1], [1,1,1,1], [1',1,1,1], and so a(4) = 6.

Links

Crossrefs

Cf. A015128, A092269, A235792, A335724, A335730.

Formula

G.f.: 2*(Product_{k>=1} (1+q^k)/(1-q^k))*Sum_{n>=1} (q^(2*n)*Product_{j=1..2*n}(1-q^j))/((1-q^(2*n))^2*Product_{j=1..2*n}(1+q^j)).
a(n) = A335724(n) - A335730(n).

A335730 a(n) is the number of smallest parts in the overpartitions of n having odd smallest part.

Original entry on oeis.org

2, 4, 12, 20, 40, 72, 124, 200, 330, 520, 804, 1224, 1832, 2704, 3960, 5704, 8144, 11532, 16164, 22480, 31056, 42568, 57972, 78480, 105610, 141336, 188208, 249352, 328824, 431760, 564468, 734992, 953424, 1232144, 1586760, 2036580, 2605352, 3322584, 4224624, 5355920
Offset: 1

Views

Author

Jeremy Lovejoy, Jun 19 2020

Keywords

Examples

			There are 14 overpartitions of 4: [4], [4'], [3,1], [3,1'], [3',1], [3',1'], [2,2], [2',2], [2,1,1], [2,1',1], [2',1,1], [2',1',1], [1,1,1,1], [1',1,1,1], and so a(4) = 20.

Links

Crossrefs

Cf. A015128, A092269, A235792, A335724, A335728.

Formula

a(n) = A335724(n) - A335728(n).
G.f.: (Product_{k>=1} (1+q^k)/(1-q^k))*(Sum_{n>=1} 2*n*q^n/(1-q^(2*n)) + Sum_{n=-oo..oo, n<>0} 4*(-1)^n*q^(n^2+n)*(1+q^(2*n)+q^(3*n))/((1-q^(2*n))*(1-q^(4*n)))).

A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 2, 4, 2, 0, 2, 6, 4, 2, 0, 2, 8, 8, 4, 2, 0, 2, 10, 14, 8, 4, 2, 0, 2, 12, 20, 16, 8, 4, 2, 0, 2, 14, 28, 26, 16, 8, 4, 2, 0, 2, 16, 38, 40, 28, 16, 8, 4, 2, 0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2, 0, 2, 20, 60, 84, 72, 48, 28, 16, 8, 4, 2
Offset: 0

Views

Author

Gregory L. Simay, Jun 14 2019

Keywords

Comments

T(n,0) = A000007(n).
T(n,1) = A040000(n) for n > 0.
T(n,2) = A005843(n-1).
T(n,3) = 2*A007980(n-3).
T(n,4) = 2*A061866(n-1).
T(n,5) = 2*A091773(n-5).
Conjecture: T(n,k) = 2*(the associated Poincaré series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincaré series.

Examples

			T(5,3) = 8 and counts the overpartitions 3,1,1; 3',1,1; 3,1',1; 3',1',1; 2,2,1; 2',2,1; 2,2,1' and 2',2,1'.
T(16,5) = 404 = T(11,5) + 2*( T(11,4) + T(11,3) + T(11,2) + T(11,1)) = 72 + 2*(84 + 60 + 20 + 2) = 404.
T(16, 5) = T(15,4) + T(11,4) + T(10,4) + T(6,4) + T(5,4) = 248 + 84 + 60 + 8 + 4 = 404.
T(9,1) + T(8,2) + T(7,3) + T(6,4) + T(6,5)= 2 + 14 + 20 + 8 + 2 = 46 =A300415(10).
Triangle: T(n,k) begins:
  1;
  0, 2;
  0, 2,  2;
  0, 2,  4,  2;
  0, 2,  6,  4,  2;
  0, 2,  8,  8,  4,  2;
  0, 2, 10, 14,  8,  4,  2;
  0, 2, 12, 20, 16,  8,  4,  2;
  0, 2, 14, 28, 26, 16,  8,  4, 2;
  0, 2, 16, 38, 40, 28, 16,  8, 4, 2;
  0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2;
  ...

Crossrefs

Row sums give A015128.
Main diagonal T(n,n) gives A040000.
Cf. A005843, A007980, A061866, A091773, A235792, A300415.

Programs

  • Maple
    b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      expand(`if`(j>0, 2*x^j, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jun 15 2019
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[If[j > 0, 2*x^j, 1]*b[n - i*j, i - 1]], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

Formula

Sum_{k=0..n} T(n,k) = A015128(n), the number of overpartitions of n.
If k > n, T(n,k) = 0.
If n >= k > n/2, T(n,k) = 2*A015128(n-k).
Conjecture: T(n,k) = T(n-k, k) + 2*(T(n-k, k-1) + ... + T(n-k, 1)).
Conjecture: T(n,k) = T(n-1, k-1) + T(n-k, k-1) + T(n-k-1, k-1) + T(n-2k, k-1) + T(n-2k-1) + ...
Conjecture: T(n,1) + T(n-1,2) + ... + T(n-floor(n/2),floor(n/2)) = A300415(n+1).
T(n,2) = 2n - 2.
Conjecture: g.f. T(n,k) = 2*(1+x)(1+x^2)...(1+x^(k-1))/((1-x)...(1-x^k)).
Sum_{k=1..n} k * T(n,k) = A235792(n). - Alois P. Heinz, Jun 15 2019
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.